# The summation of infinite partial fraction decomposition I: some formulae
related to the Hurwitz zeta function

Xiaowei Wang
This paper is written in Dec 2020

###### Abstract

In this paper we establish a new summation method by expanding
$\prod_{k}(1-\frac{z}{a_{k}})^{-1}$ with two approaches: the Taylor expansion
and the infinite partial fraction decomposition. Here we focus on the case
when $a_{k}$ is arithmetic sequence. By this summation we obtain many
equalities involve Hurwitz zeta function and Gammma function.

_K_ eywords Partial fraction, Summation, Hurwitz zeta function

##  1 Introduction

Let $S$ be the series of rational functions $S=\sum\frac{P(n)}{Q(n)}$ with
$\deg(Q)\geq\deg(P)+2$. By decomposing $\frac{P}{Q}$ into partial fractions,
$S$ can be rewritten as finite linear combination of Hurwitz zeta function
$\zeta(m,a)=\sum_{k=0}^{\infty}\frac{1}{(a+k)^{m}}$ with positive interger
$m$. In this paper, we construct another series representation of $\zeta(m,a)$
by discuss two kinds of expansions of the infinite product
$\prod_{k=0}^{\infty}(1-(\frac{z}{a+k})^{m})$.
In fact, the correspondence is natural. In general, there is some relation
between $\sum_{k=0}^{\infty}\frac{1}{a_{k}^{m}}$ and
$\sum_{k=0}^{\infty}\frac{1}{F^{\prime}(a_{k})}\frac{1}{a_{k}^{m+1}}$, where
$F(z)$ is the product $\prod_{k=0}^{\infty}(1-(\frac{z}{a_{k}})^{m})$.
Basically, the relation is derived by the infinite partial fraction
decomposition of $\frac{1}{F(z)}$. And the correspondence is not only valid
for infinite series, but also valid for finite sum. Therefore the convergence
can be discussed easily. In this we paper we study this relation and focus on
the case when $a_{k}=a+k$, namely, the problem involves the Hurwitz zeta
function.
Throughout the paper we denote the coefficient of $z^{n}$ in the Laurent
expansion of $f(z)$ around $0$ by $[f(z)]_{n}$. The notation
$F^{\prime}(a_{n})$ refers to $\lim_{z\rightarrow a_{n}}F^{\prime}(z)$.

##  2 Prelimilary

######  Lemma 1.

(Homogeneous partial fraction decomposition)
Let $a_{1},...,a_{n}$ be distinct complex numbers,
$x\in\mathbb{C}\backslash\\{a_{1},...,a_{n}\\}$, then there exist
$\mu_{1},...,\mu_{n}\in\mathbb{C}$ such that following identity is true,

$\prod_{i=1}^{n}\frac{1}{x-a_{i}}=\sum_{i=1}^{n}\frac{\mu_{i}}{x-a_{i}}$ (1)

where $\mu_{i}=\prod_{j=1,j\neq i}^{n}\frac{1}{a_{i}-a_{j}}$. Further, The
relation $\sum_{i=1}^{n}\mu_{i}=0$ holds.

###### Proof.

The proof is easy, one can consider the Lagrange’s interpolation formula. For
more details, See author’s paper [1]. ∎

######  Lemma 2.

(One point lifting)
Suppose that $N,L\in\mathbb{Z}_{\geq 0}$ and $a_{n}$ are complex numbers that
differ from each other, then the following identity holds

$\frac{1}{x^{L}}\prod_{n=1}^{N}\frac{1}{x-a_{n}}=\sum_{j=1}^{L}\sum_{n=1}^{N}\frac{-\mu_{n}}{a_{n}^{L-j+1}}\frac{1}{x^{j}}+\sum_{n=1}^{N}\frac{\mu_{n}}{a_{n}^{L}}\frac{1}{x-a_{n}}$

where

$\mu_{n}=\prod_{s=1,s\neq n}^{N}\frac{1}{a_{n}-a_{s}}$

###### Proof.

Firstly one can see the case $L=0$ is exactly Lemma 1, hence the the identity
is true for $\ell=0$. Following we assume that $L\in\mathbb{Z}_{\geq 1}$.
Suppose that $z_{0},z_{1},...,z_{N}$ be complex variables that take distinct
values. Let $H(z_{0},z_{1},...,z_{n})=\prod_{n=0}^{N}\frac{1}{x-z_{n}}$. Then
by Lemma 1 we have the identity

$H(z_{0},z_{1},...,z_{n})=\sum_{n=0}^{N}\frac{\lambda_{n}}{x-z_{n}}$

Where $\lambda_{n}=\prod_{s=0,s\neq n}^{N}\frac{1}{z_{n}-z_{s}}$. Now we
taking partial derivatives of $H$ with respect to $z_{0}$. There are two ways
to carry out. On the one hand,

$\frac{\partial^{l}H}{\partial
z_{0}^{l}}=\frac{l!}{(x-z_{0})^{l+1}}\prod_{n=1}^{\infty}\frac{1}{x-z_{n}}$
(2)

On the other hand,

$\displaystyle\frac{\partial^{l}H}{\partial z_{0}^{l}}$
$\displaystyle=\frac{\partial^{l}}{\partial
z_{0}^{l}}(\lambda_{0}\frac{1}{x-z_{0}})+\sum_{n=1}^{N}\frac{1}{x-z_{n}}\frac{\partial^{l}\lambda_{n}}{\partial
z_{0}^{l}}$
$\displaystyle=\sum_{j=0}^{l}\binom{l}{j}\frac{\partial^{l-j}\lambda_{0}}{\partial
z_{0}^{l-j}}\frac{j!}{(x-z_{0})^{j+1}}+\sum_{n=1}^{N}\frac{1}{x-z_{n}}\frac{\partial^{l}\lambda_{n}}{\partial
z_{0}^{l}}$

Recall that $\lambda_{0}=\prod_{s=1}^{N}\frac{1}{z_{0}-z_{s}}$. reapplying
Lemma 1

$\lambda_{0}=\sum_{n=1}^{N}\frac{M_{n}}{z_{0}-z_{n}}$

where $M_{n}=\prod_{s=1,s\neq n}^{N}\frac{1}{z_{n}-z_{s}}$. Therefore

$\frac{\partial^{l-j}\lambda_{0}}{\partial
z_{0}^{l-j}}=\sum_{n=1}^{N}M_{n}\frac{(-1)^{l-j}(l-j)!}{(z_{0}-z_{n})^{l-j+1}}$

Further, for $n=1,...,N$

$\frac{\partial^{l}\lambda_{n}}{\partial
z_{0}^{l}}=\frac{l!M_{n}}{(z_{n}-z_{0})^{l+1}}$

Now compare to (2) we obtain

$\frac{1}{(x-z_{0})^{l+1}}\prod_{n=1}^{\infty}\frac{1}{x-z_{n}}=\sum_{j=0}^{l}\sum_{n=1}^{N}\frac{M_{n}(-1)^{l-j}(l-j)!}{(z_{0}-z_{n})^{l-j+1}}\frac{1}{(x-z_{0})^{j+1}}+\sum_{n=1}^{N}\frac{1}{x-z_{n}}\frac{M_{n}}{(z_{n}-z_{0})^{l+1}}$

Lastly let $L=l+1$, $z_{0}=0$, and $z_{i}=a_{i}$,
$\mu_{n}=M_{n}|_{z_{i}=a_{i}}$ for $i=1,...,N$ respectively, we obtain what
required. ∎

##  3 Main Results

In the last section we introduce some lemmas about partial fraction
decomposition. In follow we discuss the partial fraction expansion of the
infinite product
$(z^{L}F(z))^{-1}=z^{-L}\prod_{n=1}^{\infty}(1-\frac{z}{a_{n}})^{-1}$. We then
show that the rearrangement to the power series of $z$ remains invariant for
different $L$.

######  Theorem 1.

(Partial Fraction Summation)
Let $(a_{n})$ be a sequence in $\mathbb{C}\backslash\\{0\\}$ and satisfy that:
1, $a_{n}$ differ from each other for all $n$; 2,
$\sum_{n=1}^{\infty}\frac{1}{a_{n}}$ absolutely converges. Define

$F(z)=\prod_{n=1}^{\infty}(1-\frac{z}{a_{n}})$

then following identity holds for $k=0,1,2,...$.

$[\frac{1}{F(z)}]_{k}=\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(a_{n})}\frac{1}{a_{n}^{k+1}}$

On the other hand, for $L=0,1,2,...$, $\frac{1}{F(z)}$ has the following
partial fraction decompositions

$\frac{1}{F(z)}=[\frac{1}{F(z)}]_{0}+[\frac{1}{F(z)}]_{1}z+...+[\frac{1}{F(z)}]_{L-1}z^{L-1}+\sum_{n=1}^{\infty}\frac{z^{L}}{F^{\prime}(a_{n})a_{n}^{L}(z-a_{n})}$

For given $L$, we call $L$ the order of this expansion.

###### Proof.

Let $M=\inf(|a_{n}|)$, in the proof we always assume that $|z|<M$. Consider
the expansion of $\frac{1}{F(z)}$, namely,

$\frac{1}{F(z)}=\exp(-\sum_{n=1}^{\infty}\log(1-\frac{z}{a_{n}}))=\exp(\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{a_{n}^{k}}\frac{z^{k}}{k})$

Denote $[\frac{1}{F(z)}]_{k}$ by $c_{k}$, it can be computed by expanding the
right hand side in the above formula. Then

$\frac{1}{z^{L}F(z)}=\frac{1}{z^{L}}(c_{0}+c_{1}z+c_{2}z^{2}+...)$ (3)

On the other hand, consider the another expansion via Lemma 2 and let
$N\rightarrow\infty$, that is

$\frac{1}{z^{L}F(z)}=\frac{1}{z^{L}}(\sum_{n=1}^{\infty}\frac{-\lambda_{n}}{a_{n}})+...+\frac{1}{z^{2}}(\sum_{n=1}^{\infty}\frac{-\lambda_{n}}{a_{n}^{L-1}})+\frac{1}{z}(\sum_{n=1}^{\infty}\frac{-\lambda_{n}}{a_{n}^{L}})+\sum_{n=1}^{\infty}\frac{\lambda_{n}}{a_{n}^{L}(z-a_{n})}$
(4)

Where $\lambda_{n}=-a_{n}\prod_{s=1,s\neq
n}^{\infty}\frac{a_{s}}{a_{s}-a_{n}}$. Further, we can reformulate
$\lambda_{n}$ as following

$\lambda_{n}=-a_{n}\lim_{z-a_{n}\rightarrow
0}\frac{a_{n}-z}{a_{n}}\prod_{s=1}^{\infty}\frac{a_{s}}{a_{s}-z}=\frac{1}{F^{\prime}(a_{n})}$

Now consider the expansion of
$\sum_{n=1}^{\infty}\frac{\lambda_{n}}{a_{n}^{L}(z-a_{n})}$, due to the
absolute convergence of $\sum\frac{1}{a_{n}}$, for $|z|<M$ we have

$\sum_{n=1}^{\infty}\frac{\lambda_{n}}{a_{n}^{L}(z-a_{n})}=\sum_{n=1}^{\infty}\frac{-\lambda_{n}}{a_{n}^{L+1}}+\sum_{n=1}^{\infty}\frac{-\lambda_{n}}{a_{n}^{L+2}}z+\sum_{n=1}^{\infty}\frac{-\lambda_{n}}{a_{n}^{L+3}}z^{2}+...$

Then equation (4) becomes

$\frac{1}{z^{L}F(z)}=\frac{1}{z^{L}}(\sum_{n=1}^{\infty}\frac{-\lambda_{n}}{a_{n}}+\sum_{n=1}^{\infty}\frac{-\lambda_{n}}{a_{n}^{2}}z+...+\sum_{n=1}^{\infty}\frac{-\lambda_{n}}{a_{n}^{L+1}}z^{L}+...)$
(5)

Compare to (3) we have for $k=0,1,2,3,...$, the equalities hold

$c_{k}=\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(a_{n})}\frac{1}{a_{n}^{k+1}}$

∎

######  Corollary 1.

As a direct corollary of theorem 1,

$\displaystyle\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(a_{n})}\frac{1}{a_{n}}\equiv
1$
$\displaystyle\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(a_{n})}\frac{1}{a_{n}^{2}}=\sum_{n=1}^{\infty}\frac{1}{a_{n}}$
$\displaystyle\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(a_{n})}\frac{1}{a_{n}^{3}}=\frac{1}{2}((\sum_{n=1}^{\infty}\frac{1}{a_{n}})^{2}+\sum_{n=1}^{\infty}\frac{1}{a_{n}^{2}})$
$\displaystyle...$

######  Theorem 2.

(Partial fraction summation for symmetric products)
Let $(a_{n})$ be a sequence in $\mathbb{C}\backslash\\{0\\}$ and satisfy that:
1, $a_{n}$ differ from each other for all $n$; 2,
$\sum_{n=1}^{\infty}\frac{1}{a_{n}^{2}}$ absolutely converges. Define

$F(z)=\prod_{n=1}^{\infty}(1-(\frac{z}{a_{n}})^{2})$

then following identity holds for $k=0,1,2,...$.

$[\frac{1}{F(z)}]_{2k}=\sum_{n=1}^{\infty}\frac{-2}{F^{\prime}(a_{n})}\frac{1}{a_{n}^{2k+1}}$

On the other hand, for $L=0,2,4,...$, $\frac{1}{F(z)}$ has the following
partial fraction decompositions

$\frac{1}{F(z)}=[\frac{1}{F(z)}]_{0}+[\frac{1}{F(z)}]_{2}z^{2}+...+[\frac{1}{F(z)}]_{L-2}z^{L-2}+\sum_{n=1}^{\infty}\frac{2z^{L}}{F^{\prime}(a_{n})a_{n}^{L-1}(z^{2}-a_{n}^{2})}$
(6)

for $L=1,3,5,...$, $\frac{1}{F(z)}$ has the following partial fraction
decompositions

$\frac{1}{F(z)}=[\frac{1}{F(z)}]_{0}+[\frac{1}{F(z)}]_{2}z^{2}+...+[\frac{1}{F(z)}]_{L-1}z^{L-1}+\sum_{n=1}^{\infty}\frac{2z^{L+1}}{F^{\prime}(a_{n})a_{n}^{L}(z^{2}-a_{n}^{2})}$
(7)

###### Proof.

By Theorem 1, one can let $F(z)=\prod_{n=1}^{\infty}(1-\frac{z}{b_{n}})$ where
$b_{2n-1}=a_{n},b_{2n}=-a_{n}$. In order the find $F^{\prime}(b_{n})$, just
need to note that $F(z)=f(z)f(-z)$, where
$f(z)=\prod_{n=1}^{\infty}(1-\frac{z}{a_{n}})$ ( This infinite product may be
divergent, one can use finite product to approximate ), Then

$\displaystyle
F^{\prime}(b_{2n-1})=(f(z)f(-z))^{\prime}|_{z=a_{n}}=f^{\prime}(a_{n})f(-a_{n})-f(a_{n})f^{\prime}(-a_{n})=f^{\prime}(a_{n})f(-a_{n})$
$\displaystyle
F^{\prime}(b_{2n})=(f(z)f(-z))^{\prime}|_{z=-a_{n}}=f^{\prime}(-a_{n})f(a_{n})-f(-a_{n})f^{\prime}(a_{n})=-f^{\prime}(a_{n})f(-a_{n})$

That is $F^{\prime}(b_{2n-1})=-F^{\prime}(b_{2n})$. By Theorem 1 we have

$\displaystyle[\frac{1}{F(z)}]_{2k-1}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(b_{n})}\frac{1}{b_{n}^{2k}}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(b_{2n-1})}\frac{1}{b_{2n-1}^{2k}}+\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(b_{2n})}\frac{1}{b_{2n}^{2k}}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(b_{2n-1})}\frac{1}{a_{n}^{2k}}+\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(b_{2n})}\frac{1}{a_{n}^{2k}}$
$\displaystyle=0$

This is trivial since $F(z)$ is even function.

$\displaystyle[\frac{1}{F(z)}]_{2k}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(b_{n})}\frac{1}{b_{n}^{2k+1}}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(b_{2n-1})}\frac{1}{b_{2n-1}^{2k+1}}+\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(b_{2n})}\frac{1}{b_{2n}^{2k+1}}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(b_{2n-1})}\frac{1}{a_{n}^{2k+1}}-\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(b_{2n})}\frac{1}{a_{n}^{2k+1}}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{-2}{F^{\prime}(a_{n})}\frac{1}{a_{n}^{2k+1}}$

The remaining partial fraction decomposition formulae (6), (7) are
straightforward, we omit the computation. ∎

Let $a\in\mathbb{C}\backslash\\{0,-1,-2,...\\}$, define

$g_{a}(z):=\frac{\Gamma(a+z)\Gamma(a-z)}{\Gamma(a)^{2}}$

Then $g_{a}(z)$ has the infinite product representation

$g_{a}(z)=\prod_{k=0}^{\infty}(1-(\frac{z}{a+k})^{2})^{-1}$ (8)

It can be proved easily by using the formula

$\frac{\Gamma(a+z)}{\Gamma(a)}=\frac{ae^{-\gamma
z}}{a+z}\prod_{n=1}^{\infty}(1+\frac{z}{a+n})^{-1}e^{\frac{z}{n}}$ (9)

Now by Theorem 2 we have the partial fraction decomposition of $g_{a}(z)$ of
order $1$

$\Gamma(a+z)\Gamma(a-z)=\Gamma(a)^{2}+\sum_{k=0}^{\infty}\frac{2(-1)^{k+1}\Gamma(2a+k)}{(a+k)k!}\frac{z^{2}}{z^{2}-(a+k)^{2}}$
(10)

In order to keep the convergence, we shall keep $\Re{(a)}<2$ and $|z|<|a|$.
Further, it’s easy to obtain $[\Gamma(a+z)\Gamma(a-z)]_{2}$ by (8), namely
$[\Gamma(a+z)\Gamma(a-z)]_{2}=\zeta(2,a)$. On the other hand, by finding the
Taylor expansion of (10), one has

######  Corollary 2.

(The PFS representation of $\zeta(2,a)$)

$\zeta(2,a)=\sum_{k=0}^{\infty}\frac{1}{(a+k)^{2}}=\sum_{k=0}^{\infty}\frac{2(-1)^{k}\Gamma(2a+k)}{\Gamma(a)^{2}k!(a+k)^{3}},(\Re(a)<2,a\neq
0,-\frac{1}{2},-1,-\frac{3}{2},...)$ (11)

######  Theorem 3.

(Partial fraction summation for cyclotomic product)
Let $(a_{n})$ be a sequence in $\mathbb{C}\backslash\\{0\\}$ and satisfy that:
I, $a_{n}$ differ from each other for all $n$;
II, $\sum_{n=1}^{\infty}\frac{1}{a_{n}^{m}}$ absolutely converges.
Define

$F(z)=\prod_{n=1}^{\infty}(1-(\frac{z}{a_{n}})^{m})$

then following identity holds for $J=0,1,2,...$.

$[\frac{1}{F(z)}]_{J}=\begin{cases}0,\text{ if }\mod(J,m)\neq 0\\\
\sum_{n=1}^{\infty}\frac{-m}{F^{\prime}(a_{n})}\frac{1}{a_{n}^{J+1}},\text{ if
}\mod(J,m)=0\\\ \end{cases}$ (12)

###### Proof.

The proof is quite similar to Theorem 2. For $k=1,2,..$, let $k=m(n-1)+r$ and

$b_{k}=b_{m(n-1)+r}=a_{n}\omega_{m}^{r-1},r=1,2,...,m$

where $\omega_{m}=e^{\frac{2i\pi}{m}}$ is the $m-$th root of unity. Then

$F(z)=\prod_{k=1}^{\infty}(1-\frac{z}{b_{k}})$

On the othe hand, let

$f_{r,N}(z)=\prod_{n=1}^{N}(1-\frac{z}{a_{n}\omega_{m}^{r-1}}),r=1,2,...,m$

Then $f_{r,N}(z)=f_{1,N}(z/\omega_{m}^{r-1})$, we denote $f_{1,N}(.)$ directly
by $f_{N}(.)$. Then $f_{r,N}(a_{n}\omega_{m}^{r-1})=f_{N}(a_{n})=0$ for $n\leq
N$. To compute $F^{\prime}(b_{k})=F^{\prime}(b_{m(n-1)+r})$:

$\displaystyle F^{\prime}(b_{m(n-1)+r})$
$\displaystyle=\lim_{N\rightarrow\infty}(f_{1,N}(z)f_{2,N}(z)...f_{m,N}(z))^{\prime}|_{z=a_{n}\omega_{m}^{r-1}}$
$\displaystyle=\lim_{N\rightarrow\infty}f_{N}(z)f_{N}(\frac{z}{\omega_{m}})...f_{N}(\frac{z}{\omega_{m}^{m-1}})(\frac{f_{N}^{\prime}(z)}{f_{N}(z)}+\frac{1}{\omega_{m}}\frac{f_{N}^{\prime}(z/\omega_{m})}{f_{N}(z/\omega_{m})}+...+\frac{1}{\omega_{m}^{m-1}}\frac{f_{N}^{\prime}(z/\omega_{m}^{m-1})}{f_{N}(z/\omega_{m}^{m-1})})|_{z=a_{n}\omega_{m}^{r-1}}$
$\displaystyle=\frac{1}{\omega_{m}^{r-1}}f^{\prime}(a_{n})f(a_{n}\omega_{m})...f(a_{n}\omega_{m}^{m-1})$

One can implies that for all $n=1,2,3,...$ and $r=1,2,...,m$,

$F^{\prime}(b_{m(n-1)+r})=\frac{1}{\omega_{m}^{r-1}}F^{\prime}(b_{m(n-1)+1})$

According to Theorem 1, we have

$\displaystyle[\frac{1}{F(z)}]_{J}$
$\displaystyle=\sum_{k=1}^{\infty}\frac{-1}{F^{\prime}(b_{k})}\frac{1}{b_{k}^{J+1}}$
$\displaystyle=\sum_{r=1}^{m}\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(b_{m(n-1)+r})}\frac{1}{(a_{n}\omega_{m}^{r-1})^{J+1}}$
$\displaystyle=\sum_{r=1}^{m}\frac{1}{\omega_{m}^{J(r-1)}}\sum_{n=1}^{\infty}\frac{-1}{F^{\prime}(b_{m(n-1)+1})}\frac{1}{a_{n}^{J+1}}$

Therefore

$[\frac{1}{F(z)}]_{J}=\begin{cases}0,\text{ if }\mod(J,m)\neq 0\\\
\sum_{n=1}^{\infty}\frac{-m}{F^{\prime}(a_{n})}\frac{1}{a_{n}^{J+1}},\text{ if
}\mod(J,m)=0\\\ \end{cases}$

∎

######  Corollary 3.

(The PFS representation of $\zeta(m,a)$)
For $m=2,3,4,...$, following identity holds,

$\zeta(m,a)=\sum_{k=0}^{\infty}\frac{m(-1)^{k}\prod_{r=1}^{m-1}\Gamma(a-\omega_{m}^{r}(a+k))}{\Gamma(a)^{m}k!(a+k)^{m+1}}$
(13)

###### Proof.

In the Theorem 3 for convenience let $k=n-1$, let $a_{n}=a-1+n=a+k$, then
$\frac{1}{F(z)}=\prod_{k=0}^{\infty}(1-(\frac{z}{a+k})^{m})^{-1}$, it easy to
see that $[\frac{1}{F(z)}]_{m}=\zeta(m,a)$, now according to Theorem 3, it
remains to find $F^{\prime}(a+k)$. Note that

$F(z)=\frac{\Gamma(a)^{m}}{\prod_{r=0}^{m-1}\Gamma(a-\omega_{m}^{r}z)}$

this can be easily derived by the formula (9). After some computations

$F^{\prime}(z)=\frac{\Gamma(a)^{m}}{\prod_{r=0}^{m-1}\Gamma(a-\omega_{m}^{r}z)}\sum_{r=0}^{m-1}\omega_{m}^{r}\psi(a-\omega_{m}^{r}z)$

Note that $\lim_{z\rightarrow-k}\frac{\psi(z)}{\Gamma(z)}=(-1)^{k-1}k!$, hence

$F^{\prime}(a+k)=\frac{\Gamma(a)^{m}k!(-1)^{k-1}}{\prod_{r=1}^{m-1}\Gamma(a-\omega_{m}^{r}(a+k))}$

Finally by (12) we obtain what required, namely

$\zeta(m,a)=\sum_{k=0}^{\infty}\frac{-m}{F^{\prime}(a+k)}\frac{1}{(a+k)^{m+1}}=\sum_{k=0}^{\infty}\frac{m(-1)^{k}\prod_{r=1}^{m-1}\Gamma(a-\omega_{m}^{r}(a+k))}{\Gamma(a)^{m}k!(a+k)^{m+1}}$

∎

######  Corollary 4.

(The PFS representation of $\zeta(m)$)
Let $a=1$, we have then

$\zeta(m)=\sum_{n=1}^{\infty}\frac{m(-1)^{n-1}\Gamma(1-\omega_{m}n)...\Gamma(1-\omega_{m}^{m-1}n)}{n!n^{m}}$

As a special case, after some simplification, we have a series representation
for Aprèy’s constant

$\zeta(3)=\sum_{n=1}^{\infty}\frac{3(-1)^{n-1}\Gamma(\frac{1+\sqrt{3}}{2}n)\Gamma(\frac{1-\sqrt{3}}{2}n)}{n\cdot
n!}$

######  Theorem 4.

(The partial fraction summation for general product)
Let

$F(z)=\prod_{k=0}^{\infty}\prod_{m=1}^{M}(1-\frac{z}{a_{m,k}})$

then formally we have

$[\frac{1}{F(z)}]_{\ell}=\sum_{k=0}^{\infty}\sum_{m=1}^{M}\frac{-1}{a_{m,k}^{\ell+1}F^{\prime}(a_{m,k})}$

especially, if $F(z)$ normally converges in $|z|<M$ for some $M>0$, then

$\displaystyle[\frac{1}{F(z)}]_{0}=\sum_{k=0}^{\infty}\sum_{m=1}^{M}\frac{-1}{a_{m,k}F^{\prime}(a_{m,k})}=1$
$\displaystyle[\frac{1}{F(z)}]_{1}=\sum_{k=0}^{\infty}\sum_{m=1}^{M}\frac{-1}{a_{m,k}^{2}F^{\prime}(a_{m,k})}=\sum_{k=0}^{\infty}\sum_{m=1}^{M}\frac{1}{a_{m,k}}$

###### Proof.

The proof is similar to the proof of Theorem 3. We left it to the reader.
∎

######  Example 5.

Let
$F(z)=\frac{\Gamma(\frac{1}{2})}{\Gamma(1+\frac{z}{2})\Gamma(\frac{1}{2}-\frac{z}{2})}$,
then we can reformulate $F(z)$ as

$F(z)=\prod_{k=1}^{\infty}(1-\frac{z}{a_{k}})(1-\frac{z}{b_{k}})$

where $a_{k}=2k-1$, $b_{k}=-2k$. It’s not hard to obtain

$F^{\prime}(z)=\frac{\sqrt{\pi}}{2\Gamma(1+\frac{z}{2})\Gamma(\frac{1}{2}-\frac{z}{2})}(\psi(\frac{1}{2}-\frac{z}{2})-\psi(1+\frac{z}{2}))$

It turns out that

$F^{\prime}(a_{k})=-F^{\prime}(b_{k})=\frac{\sqrt{\pi}}{2}\frac{(-1)^{k}(k-1)!}{\Gamma(k+\frac{1}{2})}$

By the Theorem 4

$\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k-1}\Gamma(k+\frac{1}{2})}{(k-1)!}(\frac{1}{2k-1}+\frac{1}{2k})=\frac{\sqrt{\pi}}{2}$
$\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k-1}\Gamma(k+\frac{1}{2})}{(k-1)!}(\frac{1}{(2k-1)^{2}}-\frac{1}{(2k)^{2}})=\frac{\sqrt{\pi}\log
2}{2}$
$\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k-1}\Gamma(k+\frac{1}{2})}{(k-1)!}(\frac{1}{(2k-1)^{3}}+\frac{1}{(2k)^{3}})=\frac{\sqrt{\pi}(\log^{2}2+\zeta(2))}{4}$

Note that
$\frac{\Gamma(k+\frac{1}{2})}{(k-1)!}=\sqrt{\pi}\frac{(2k-1)!!}{(2k-2)!!}$, in
fact

$\displaystyle\sum_{k=1}^{\infty}(-1)^{k-1}\frac{(2k-1)!!}{(2k-2)!!}(\frac{1}{2k-1}+\frac{1}{2k})=\frac{1}{2}$
$\displaystyle\sum_{k=1}^{\infty}(-1)^{k-1}\frac{(2k-1)!!}{(2k-2)!!}(\frac{1}{(2k-1)^{2}}-\frac{1}{(2k)^{2}})=\frac{\log
2}{2}$
$\displaystyle\sum_{k=1}^{\infty}(-1)^{k-1}\frac{(2k-1)!!}{(2k-2)!!}(\frac{1}{(2k-1)^{3}}+\frac{1}{(2k)^{3}})=\frac{\log^{2}2+\zeta(2)}{2}$

More general, we have

######  Example 6.

Assume that $0\leq\Re(a),\Re(b)<1$ and $a,b\neq 0$. Let $a_{k}=a+k$,
$b_{k}=-b-k$, $k=0,1,2,...$. Consider

$F(z)=\prod_{k=0}^{\infty}(1-\frac{z}{a_{k}})(1-\frac{z}{b_{k}})=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a-z)\Gamma(b+z)}$

One can show that

$F^{\prime}(z)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a-z)\Gamma(b+z)}(\psi(a-z)-\psi(b-z))$

That is

$\frac{1}{F^{\prime}(a_{k})}=-\frac{1}{F^{\prime}(b_{k})}=\frac{(-1)^{k-1}\Gamma(a+b+k)}{k!\Gamma(a)\Gamma(b)}$

Therefore

$[\frac{1}{F(z)}]_{J}=\sum_{k=0}^{\infty}\frac{(-1)^{k}\Gamma(a+b+k)}{k!\Gamma(a)\Gamma(b)}(\frac{1}{(a+k)^{J+1}}+\frac{(-1)^{m}}{(b+k)^{J+1}})$

If we consider some special cases for instance $[\frac{1}{F(z)}]_{0}$ and
$[\frac{1}{F(z)}]_{1}$, then

$\Gamma(a)\Gamma(b)=\sum_{k=0}^{\infty}\frac{(-1)^{k}\Gamma(a+b+k)}{k!}(\frac{1}{a+k}+\frac{1}{b+k})$

and

$\Gamma(a)\Gamma(b)(\sum_{k=0}^{\infty}\frac{1}{a+k}-\frac{1}{b+k})=\sum_{k=0}^{\infty}\frac{(-1)^{k}\Gamma(a+b+k)}{k!}(\frac{1}{(a+k)^{2}}-\frac{1}{(b+k)^{2}})$

Moreover, if we let $a=\frac{1}{4},b=\frac{3}{4}$, then it becomes

$\Gamma(\frac{1}{4})\Gamma(\frac{3}{4})(\sum_{k=0}^{\infty}\frac{1}{\frac{1}{4}+k}-\frac{1}{\frac{3}{4}+k})=\sum_{k=0}^{\infty}(-1)^{k}(\frac{1}{(\frac{1}{4}+k)^{2}}-\frac{1}{(\frac{3}{4}+k)^{2}})$

or

$\sum_{k=0}^{\infty}(-1)^{\frac{k(k+1)}{2}}\frac{1}{(2k+1)^{2}}=\frac{\sqrt{2}\pi^{2}}{16}$

##  4 The formula of derivative

We can take derivative with respect to $a$ in (13). As a simple application,
following we discuss the case when $m=2$.

######  Theorem 7.

$\Gamma(a+z)\Gamma(a-z)(\psi(a+z)+\psi(a-z))=2\psi(a)\Gamma(a)^{2}+S_{1}(a,z)-S_{2}(a,z)$

where

$\displaystyle
S_{1}(a,z)=\sum_{k=0}^{\infty}\sum_{n=1}^{\infty}\frac{4(-1)^{k}\psi(2a+k)\Gamma(2a+k)}{k!(a+k)^{2n+1}}z^{2n}$
$\displaystyle
S_{2}(a,z)=\sum_{k=0}^{\infty}\sum_{n=1}^{\infty}\frac{2(2n+1)(-1)^{k}\Gamma(2a+k)}{k!(a+k)^{2n+2}}z^{2n}$

###### Proof.

The proof is easy, recall (10)

$\Gamma(a+z)\Gamma(a-z)=\Gamma(a)^{2}+\sum_{k=0}^{\infty}\frac{2(-1)^{k+1}\Gamma(2a+k)}{(a+k)k!}\frac{z^{2}}{z^{2}-(a+k)^{2}}$

where it converges in $|z|<a$. The right hand side
$\frac{z^{2}}{z^{2}-(a+k)^{2}}$ can be represented as
$-\sum_{n=1}^{\infty}(\frac{z}{a+k})^{2n}$. It can be reformulated as

$\Gamma(a+z)\Gamma(a-z)=\Gamma(a)^{2}+\sum_{k=0}^{\infty}\sum_{n=1}^{\infty}\frac{2(-1)^{k}\Gamma(2a+k)}{k!(a+k)^{2n+1}}z^{2n}$

Take $\frac{\partial}{\partial a}$ on both sides then obtain what required.
∎

######  Corollary 5.

Denote

$\displaystyle\zeta_{A}(s)=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^{s}}$
$\displaystyle\zeta_{A,H}(s)=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}H_{k}}{k^{s}}$

where $H_{k}$ is the harmonic number $H_{k}=1+\frac{1}{2}+...+\frac{1}{k}$.
Then the following relation is valid for $n\in\mathbb{Z}^{+}$.

$\zeta_{A,H}(2n)=(n-\frac{n}{2^{2n}}-\frac{1}{2^{2n+1}})\zeta(2n+1)-\sum_{j=1}^{n-1}\zeta(2j+1)\zeta_{A}(2n-2j)$

###### Proof.

Let $a=1$ then according to Theorem 7

$\frac{\pi z}{\sin(\pi
z)}(\psi(1+z)+\psi(1-z))=-2\gamma+S_{1}(1,z)-S_{2}(1,z)$ $\displaystyle
S_{1}(1,z)$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{n=1}^{\infty}\frac{4(-1)^{k}\psi(2+k)}{(1+k)^{2n}}z^{2n}$
$\displaystyle=-\gamma\sum_{k=0}^{\infty}\sum_{n=1}^{\infty}\frac{4(-1)^{k}}{(1+k)^{2n}}z^{2n}+\sum_{k=0}^{\infty}\sum_{n=1}^{\infty}\frac{4(-1)^{k}H_{1+k}}{(1+k)^{2n}}z^{2n}$

or rewritten as

$[S_{1}(1,z)]_{2n}=-4\gamma\zeta_{A}(2n)+4\zeta_{A,H}(2n),(n\geq 1)$

On the other hand

$\displaystyle S_{2}(a,z)$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{n=1}^{\infty}\frac{2(2n+1)(-1)^{k}\Gamma(2a+k)}{k!(a+k)^{2n+2}}z^{2n}$
$\displaystyle=\sum_{n=1}^{\infty}\sum_{k=0}^{\infty}\frac{2(2n+1)(-1)^{k}}{(k+1)^{2n+1}}z^{2n}$

or rewritten as

$[S_{2}(1,z)]_{2n}=2(2n+1)\zeta_{A}(2n+1),(n\geq 1)$

Therefore

$[S_{1}(1,z)-S_{2}(1,z)]_{2n}=-4\gamma\zeta_{A}(2n)+4\zeta_{A,H}(2n)-2(2n+1)\zeta_{A}(2n+1),(n\geq
1)$

If we denote $[\frac{\pi z}{\sin(\pi z)}]_{n}$ by $\tau_{n}$, and note that
$(\psi(1+z)+\psi(1-z))=-2(\gamma+\sum_{n=1}^{\infty}\zeta(2n+1)z^{2n})$ then

$[\frac{\pi z}{\sin(\pi
z)}(\psi(1+z)+\psi(1-z))]_{2n}=-2(\gamma\tau_{2n}+\zeta(2n+1)+\sum_{j=1}^{n-1}\zeta(2j+1)\tau_{2n-2j})$

Therefore

$2\gamma\zeta_{A}(2n)-2\zeta_{A,H}(2n)+(2n+1)\zeta_{A}(2n+1)=\gamma\tau_{2n}+\zeta(2n+1)+\sum_{j=1}^{n-1}\zeta(2j+1)\tau_{2n-2j}$

One can show that $\tau_{n}=2\zeta_{A}(2n)$ and
$\zeta_{A}(s)=(1-\frac{1}{2^{s-1}})\zeta(s)$. It follows that

$\zeta_{A,H}(2n)=(n-\frac{n}{2^{2n}}-\frac{1}{2^{2n+1}})\zeta(2n+1)-\sum_{j=1}^{n-1}\zeta(2j+1)\zeta_{A}(2n-2j)$

∎

######  Corollary 6.

Denote

$\displaystyle\beta(s)=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k-1)^{s}}$
$\displaystyle\beta_{H}(s)=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}H_{k-1}}{(2k-1)^{s}}$

Then for $n\in\mathbb{Z}^{+}$ the following relation is valid.

$\beta_{H}(2n+1)=(2n+1)\beta(2n+2)-\beta(2n+1)\log
4-2\sum_{j=0}^{n-1}(1-\frac{1}{2^{2n-2j+1}})\beta(2j+1)\zeta(2n-2j+1)$

###### Proof.

Let $u_{2n}=[\psi(\frac{1}{2}+z)+\psi(\frac{1}{2}-z)]_{2n}$, one can show that

$u_{2n}=\begin{cases}2(1-2^{2n+1})\zeta(2n+1),\text{if }n\geq 1\\\
2(-\gamma-\log 4),\text{if }n=0\end{cases}$

On the other hand,

$\frac{\pi}{\cos(\pi
z)}=\sum_{k=0}^{\infty}\frac{(-1)^{k}E_{2k}\pi^{2k+1}}{(2k)!}z^{2k}$

where $E_{2k}$ is Euler number $E_{0}=1,E_{2}=-1,E_{4}=5,E_{6}=-61,...$.
Recall the formula about the Dirichlet beta function
$\beta(2n+1)=\frac{(-1)^{n}E_{2n}\pi^{2n+1}}{2^{2n+2}(2n)!}$ we have

$[\frac{\pi}{\cos(\pi z)}]_{2n}=2^{2n+2}\beta(2n+1)$

One can show that

$\displaystyle
S_{1}(\frac{1}{2},z)=\sum_{k=0}^{\infty}\sum_{n=1}^{\infty}\frac{4(-1)^{k}\psi(1+k)}{(\frac{1}{2}+k)^{2n+1}}z^{2n}$
$\displaystyle
S_{2}(\frac{1}{2},z)=\sum_{k=0}^{\infty}\sum_{n=1}^{\infty}\frac{2(2n+1)(-1)^{k}}{(\frac{1}{2}+k)^{2n+2}}z^{2n}$

or rewritten as

$\displaystyle[S_{1}(\frac{1}{2},z)]_{2n}=2^{2n+3}(-\gamma\beta(2n+1)+\beta_{H}(2n+1))$
$\displaystyle[S_{2}(\frac{1}{2},z)]_{2n}=2^{2n+3}(2n+1)\beta(2n+2)$

According to Theorem 7, by some computation one can get

$\beta_{H}(2n+1)=(2n+1)\beta(2n+2)-\beta(2n+1)\log
4-2\sum_{j=0}^{n-1}(1-\frac{1}{2^{2n-2j+1}})\beta(2j+1)\zeta(2n-2j+1)$

∎

##  5 Differential Relation

######  Theorem 8.

(Differential Relation)
Suppose that the sequence $(a_{k})$ satisfies the condictions in Theorem 2.
Let $F(z)=z\prod_{k=0}^{\infty}(1-(\frac{z}{a_{k}})^{2})$,
$H(z)=\frac{F^{\prime}(z)}{F(z)}$, then

$\frac{F^{\prime\prime}(z)}{F(z)}=H^{\prime}(z)+H^{2}(z)=\frac{F^{\prime\prime}(\infty)}{F(\infty)}+\sum_{k=0}^{\infty}\frac{F^{\prime\prime}(a_{k})}{F^{\prime}(a_{k})}\frac{2a_{k}}{z^{2}-a_{k}^{2}}$

where

$\frac{F^{\prime\prime}(\infty)}{F(\infty)}=\sum_{k=0}^{\infty}\frac{2F^{\prime\prime}(a_{k})}{a_{k}F^{\prime}(a_{k})}-6\sum_{k=0}^{\infty}\frac{1}{a_{k}^{2}}$

Equivalently, the relation can be reformulated as

$[\frac{F^{\prime\prime}(z)}{F(z)}]_{J}=\begin{cases}0,\text{ if }J\text{
odd}\\\ -6\sum_{k=0}^{\infty}\frac{1}{a_{k}2},\text{ if }J=0\\\
\sum_{k=0}^{\infty}\frac{-2F^{\prime\prime}(a_{k})}{a_{k}^{J+1}F^{\prime}(a_{k})},\text{
if }J=2,4,6,...\end{cases}$

###### Proof.

Note that $H=(\log F(z))^{\prime}$. It turns out that

$H(z)=\frac{1}{z}+\sum_{k=0}^{\infty}\frac{2z}{z^{2}-a_{k}^{2}}$ (14)

For convenience define $\widetilde{H}(z):=H(z)-\frac{1}{z}$, by the
convergence of $\sum\frac{1}{|a_{k}|^{2}}$ one can see that $\widetilde{H}(z)$
converges normally in $|z|<M$ By some easy computation one has

$\widetilde{H}^{\prime}(z)=\frac{1}{z}\widetilde{H}(z)-\sum_{k=0}^{\infty}(\frac{2z}{z^{2}-a_{k}^{2}})^{2}$

On the other hand, by (14)

$\widetilde{H}(z)^{2}=\sum_{k=0}^{\infty}(\frac{2z}{z^{2}-a_{k}^{2}})^{2}+\sum_{i\neq
j,i,j\geq 0}\frac{2z}{z^{2}-a_{i}^{2}}\frac{2z}{z^{2}-a_{j}^{2}}$

Therefore the following relation is valid

$\widetilde{H}^{\prime}(z)+\widetilde{H}(z)^{2}-\frac{1}{z}\widetilde{H}(z)=\Delta$

where

$\Delta=\sum_{i\neq j,i,j\geq
0}\frac{2z}{z^{2}-a_{i}^{2}}\frac{2z}{z^{2}-a_{j}^{2}}$

Taking the partial fraction decomposition, then

$\frac{\Delta}{4z^{2}}=\sum_{k=0}^{\infty}\widetilde{\delta}_{k}\frac{1}{z^{2}-a_{k}^{2}}$

where

$a_{k}\widetilde{\delta}_{k}=\sum_{s\neq
k,s=0}^{\infty}\frac{2a_{k}}{a_{k}^{2}-a_{s}^{2}}=\lim_{z\rightarrow
a_{k}}(\widetilde{H}(z)-\frac{2z}{z^{2}-a_{k}^{2}})=\frac{F^{\prime\prime}(a_{k})}{2F^{\prime}(a_{k})}-\frac{3}{2a_{k}}$

Hence

$\widetilde{H}^{\prime}(z)+\widetilde{H}(z)^{2}-\frac{1}{z}\widetilde{H}(z)=\sum_{k=0}^{\infty}(\frac{F^{\prime\prime}(a_{k})}{2a_{k}F^{\prime}(a_{k})}-\frac{3}{2a_{k}^{2}})\frac{4z^{2}}{z^{2}-a_{k}^{2}}$

or reformulated as

$\frac{F^{\prime\prime}(z)}{F(z)}-\frac{3F^{\prime}(z)}{zF(z)}+\frac{3}{z^{2}}=\sum_{k=0}^{\infty}(\frac{F^{\prime\prime}(a_{k})}{2a_{k}F^{\prime}(a_{k})}-\frac{3}{2a_{k}^{2}})\frac{4z^{2}}{z^{2}-a_{k}^{2}}$

Moreover, after some simplification, one has a more beautiful relation

$\frac{F^{\prime\prime}(z)}{F(z)}=\frac{F^{\prime\prime}(\infty)}{F(\infty)}+\sum_{k=0}^{\infty}\frac{F^{\prime\prime}(a_{k})}{F^{\prime}(a_{k})}\frac{2a_{k}}{z^{2}-a_{k}^{2}}$
(15)

where

$\frac{F^{\prime\prime}(\infty)}{F(\infty)}=\sum_{k=0}^{\infty}\frac{2F^{\prime\prime}(a_{k})}{a_{k}F^{\prime}(a_{k})}-6\sum_{k=0}^{\infty}\frac{1}{a_{k}^{2}}$

If we expand (15) as power series around $0$, we can find that

$[\frac{F^{\prime\prime}(z)}{F(z)}]_{J}=\begin{cases}0,\text{ if }J\text{
odd}\\\ -6\sum_{k=0}^{\infty}\frac{1}{a_{k}2},\text{ if }J=0\\\
\sum_{k=0}^{\infty}\frac{-2F^{\prime\prime}(a_{k})}{a_{k}^{J+1}F^{\prime}(a_{k})},\text{
if }J=2,4,6,...\end{cases}$

∎

######  Corollary 7.

If we define $c_{m}=\sum_{k=0}^{\infty}\frac{1}{a_{k}^{2m}}$ and
$\delta_{k}=\frac{F^{\prime\prime}(a_{k})}{F^{\prime}(a_{k})}$, then the
following relation holds for $m=1,2,3,...$

$(2m+3)c_{m+1}-2\sum_{i=1}^{m}c_{i}c_{m+1-i}=\sum_{k=0}^{\infty}\frac{\delta_{k}}{a_{k}^{2m+1}}$

###### Proof.

In Theorem 8

$H^{\prime}(z)+H^{2}(z)=\frac{F^{\prime\prime}(z)}{F(z)}$

For $H(z)=\frac{1}{z}+\sum_{k=1}^{\infty}\frac{2z}{z^{2}-a_{k}^{2}}$, there is
an expansion

$H(z)=\frac{1}{z}-2\sum_{n=1}^{\infty}c_{n}z^{2n-1}$

By the rearrangement we have

$H^{\prime}(z)+H^{2}(z)=-6c_{1}+\sum_{m=1}^{\infty}\widetilde{C}_{m}z^{2m}$

where

$\widetilde{C}_{m}=-2(2m+3)c_{m+1}+4\sum_{i=1}^{m}c_{i}c_{m+1-i}$

That is

$[H^{\prime}(z)+H^{2}(z)]_{m}=\begin{cases}0,\text{ if }m\text{ odd}\\\
-6c_{1},\text{ if }m=0\\\ -2(2m+3)c_{m+1}+4\sum_{i=1}^{m}c_{i}c_{m+1-i},\text{
if }m=2,4,6,...\end{cases}$

According to Theorem 8

$[\frac{F^{\prime\prime}(z)}{F(z)}]_{J}=\begin{cases}0,\text{ if }J\text{
odd}\\\ -6\sum_{k=0}^{\infty}\frac{1}{a_{k}2},\text{ if }J=0\\\
\sum_{k=0}^{\infty}\frac{-2\delta_{k}}{a_{k}^{J+1}},\text{ if
}J=2,4,6,...\end{cases}$

By comparing the coefficients one obtain the conclusion. ∎

######  Example 9.

A special case is when $F(z)=\pi\sin(\pi z)$. It follows from
$F(z)=z\prod_{k=0}^{\infty}(1-(\frac{z}{1+k})^{2})=\pi\sin(\pi z)$ one has
$\frac{F^{\prime\prime}(z)}{F(z)}=-\pi^{2}$ where $a_{k}=1+k$. Therefore
$\delta_{k}=\frac{F^{\prime\prime}(a_{k})}{F^{\prime}(a_{k})}=0$. If we denote
$\zeta(2n)$ by $c_{n}$, then according to Corollary 7

$(2m+3)c_{m+1}-2\sum_{i=1}^{m}c_{i}c_{m+1-i}=0$

or rewritten as

$(2m+3)\zeta(2m+2)=2\sum_{i=1}^{m}\zeta(2i)\zeta(2m+2-2i)$

This is a well-known recursion relation for $\zeta(2n)$.

Following example provides a similar relation for $\zeta(2n,a)$

######  Example 10.

Let
$F(z)=z\prod_{k=0}^{\infty}(1-(\frac{x}{a+k})^{2})=\frac{z\Gamma(a)^{2}}{\Gamma(a+z)\Gamma(a-z)}$.
It’s routine to to find $\frac{F^{\prime\prime}(z)}{F^{\prime}(z)}$. Here
$a_{k}=a+k$.

$\frac{F^{\prime\prime}(z)}{F^{\prime}(z)}=\frac{z(\psi(a-z)-\psi(a+z))^{2}+2(\psi(a-z)-\psi(a+z))-z(\psi^{\prime}(a-z)+\psi^{\prime}(a+z))}{1+z(\psi(a-z)-\psi(a+z))}$

therefore

$\displaystyle\frac{F^{\prime\prime}(a_{k})}{F^{\prime}(a_{k})}$
$\displaystyle=\frac{(a+k)(\psi(-k)-\psi(2a+k))^{2}+2(\psi(-k)-\psi(2a+k))-(a+k)(\psi^{\prime}(-k)+\psi^{\prime}(2a+k))}{1+(a+k)(\psi(-k)-\psi(2a+k))}$
$\displaystyle=\frac{(a+k)(\psi(-k)^{2}-\psi^{\prime}(-k))+(2-2(a+k)\psi(2a+k))\psi(-k)+M_{1}}{(a+k)\psi(-k)+M_{2}}$
$\displaystyle=\lim_{z\rightarrow-k}\frac{\psi(z)^{2}-\psi^{\prime}(z)}{\psi(z)}+\frac{2}{a+k}-2\psi(2a+k)$

Note that

$\lim_{z\rightarrow 0}\frac{\psi(z)^{2}-\psi^{\prime}(z)}{\psi(z)}=-2\gamma$

and recall$\psi(z+k)=\psi(z)+\frac{1}{z}+\frac{1}{z+1}+...+\frac{1}{z+k-1}$.
Hence

$\lim_{z\rightarrow-k}\frac{\psi(z)^{2}-\psi^{\prime}(z)}{\psi(z)}=2(H_{k}-\gamma)$

Therefore

$\displaystyle\delta_{k}=\frac{F^{\prime\prime}(a_{k})}{F^{\prime}(a_{k})}$
$\displaystyle=2(H_{k}+\frac{1}{a+k}-\gamma-\psi(2a+k))$
$\displaystyle=2(\psi(1+k)+\frac{1}{a+k}-\psi(2a+k))$

This makes

$\delta_{k}=\frac{F^{\prime\prime}(a_{k})}{F^{\prime}(a_{k})}=2(H_{k}+\frac{1}{a+k}-\gamma-\psi(2a+k))=2(\psi(1+k)+\frac{1}{a+k}-\psi(2a+k))$

Therefore we have the conclusion according to Corollary 7:

######  Theorem 11.

(recursion formula for $\zeta(2m,a)$) For $m=1,2,3,...$ we have

$(m+\frac{1}{2})\zeta(2m+2,a)-\sum_{i=1}^{m}\zeta(2i,a)\zeta(2m-2i+2,a)=\sum_{k=0}^{\infty}\frac{\psi(1+k)-\psi(2a+k)}{(a+k)^{2m+1}}$

Observe the formula (15), we conjecture that there is a more general relation

######  Conjecture 1.

$\frac{F^{(L)}(z)}{F(z)}=\frac{F^{(L)}(\infty)}{F(\infty)}+\sum_{k=0}^{\infty}\sum_{r=0}^{m-1}\frac{F^{(L)}(a_{k})}{F^{\prime}(a_{k})}(\frac{\omega_{m}^{r}}{z-a_{k}\omega_{m}^{r}})$

## References

  * [1] Xiaowei Wang, A new series representation involving root of unity for the values of Riemann zeta function at integer arguments, Arxiv preprint: https://arxiv.org/abs/2010.07112v2

Xiaowei Wang(王骁威)
Institut für Mathematik, Universität Potsdam, Potsdam OT Golm, Germany
Email<EMAIL_ADDRESS>


#  Behind the Deepfake: 8% Create; 90% Concerned
Surveying public exposure to and perceptions of deepfakes in the UK

Tvesha Sippy  Florence E. Enock  Jonathan Bright  Helen Z. Margetts

###### Abstract

This article examines public exposure to and perceptions of deepfakes based on
insights from a nationally representative survey of 1403 UK adults. The survey
is one of the first of its kind since recent improvements in deepfake
technology and widespread adoption of political deepfakes. The findings reveal
three key insights. First, on average, 15% of people report exposure to
harmful deepfakes, including deepfake pornography, deepfake frauds/scams and
other potentially harmful deepfakes such as those that spread health or
religious misinformation or propaganda. In terms of common targets, exposure
to deepfakes featuring celebrities was 50.2%, whereas those featuring
politicians was at 34.1%. Analysing gender and age effects, we find that men
and younger people were more likely to report being exposed to deepfakes. And
5.7% of respondents recall exposure to a selection of high profile political
deepfakes in the UK. Second, while exposure to harmful deepfakes was
relatively low, awareness of and fears about deepfakes were high (and women
were significantly more likely to report experiencing such fears than men). As
with fears, general concerns about the spread of deepfakes were also high;
90.4% of the respondents were either very concerned or somewhat concerned
about this issue. Specifically, most respondents (at least 91.8%) were
concerned that deepfakes could add to online child sexual abuse material,
increase distrust in information and manipulate public opinion. Third, while
awareness about deepfakes was high, usage of deepfake tools was relatively low
(8%). And most respondents were not confident about their ability to detect
deepfakes. At the same time, most people were trustful of audiovisual content
online. Our work highlights how the problem of deepfakes has become embedded
in public consciousness in just a few years; it also highlights the need for
media literacy programmes and other policy interventions to address the spread
of harmful deepfakes.

###### keywords:

Deepfakes , Online harm , Internet safety , Social media , Public attitudes ,
Survey research

[label1]organization=Public Policy Programme, The Alan Turing Institute,
city=London, country=UK

[label2]organization=Oxford Internet Institute, University of Oxford,
city=Oxford, country=UK

##  1 Introduction

###  1.1 The problem of deepfakes

The spread of “deepfakes”, a catch-all term referring to audio, image and
video content that has been manipulated or created using machine learning
methods to alter how a person, object or environment is presented (Centre for
Data Ethics and Innovation, 2019), is a source of increasing global concern.
Improvements in deepfake creation technology have been significant over the
last five years, making them both easier and cheaper to create, as well as
being more effective and leading some to present them as one of the most
advanced forms of visual disinformation (Weikmann et al., 2024). Their hyper-
realism makes it difficult for people to discern them from real media (Busch
and Ware, 2023). By enabling alterations to facial features and human voice
(Paris and Donovan, 2019), deepfake technologies can be deceptive on multiple
sensory levels (Brown et al., 2023), making them powerful drivers of false
narratives. Metacognitive experiences associated with audiovisual vis-a-vis
textual content may increase deepfakes’ perceived credibility (Vaccari and
Chadwick, 2020). Many fear that deepfakes could accelerate the risks of
existing misinformation: manipulating public opinion, causing political
unrest, influencing voting, encouraging harm based on conspiracies,
reinforcing incorrect beliefs, and causing distrust in information (Enock et
al., 2024). Even if deepfakes are not more deceptive than false information in
textual form (Hameleers et al., 2022), the concern that people may lose faith
in all visual material when processing information remains (Weikmann et al.,
2024).

Politically motivated deepfakes are one of the biggest areas of concern. The
deepfake audio of US President Joe Biden (Knibbs, 2024), wherein voters from
New Hampshire were discouraged from participating in the electoral process, is
one prime example. Meanwhile, in the UK, a deepfake of London’s mayor Sadiq
Khan was allegedly circulated by a far-right group in 2023; the audio is
believed to have caused a stir to local law and order (Spring, 2024). Other
politicians including Prime Minister Rishi Sunak and Sir Keir Starmer have
also been ‘deepfaked’ in the UK (Good Law Project, 2024).

Such concerns are not limited to the remit of political deepfakes. Deepfakes
are being used to perpetuate financial frauds (Bateman, 2020) and violence
against women (Kweilin, 2022). Research suggests that non-consensual
pornography constitutes 96% of deepfake videos found online and women are
disproportionately targeted in these videos (Ajder et al., 2019). In a
laudable move, the UK government recently criminalised the creation of
sexually explicit deepfakes under an amendment to the Criminal Justice Bill.
However, experts posit that the legislation falls short, to the extent that it
requires proof of perpetrator intent for prosecution (EndViolenceAgainstWomen,
2024). As with sexually explicit deepfakes, legislation for curbing political
deepfakes also exist under the Online Safety Act 2023 and the Representation
of People Act 1983. However, as experts from the Good Law Project (2024) note,
clear regulatory guidance for enforcement is needed.

However, despite the concern, there is a shortage of recent evidence on the
prevalence or impacts of deepfakes, with much existing work (that we review
fully below) predating recent improvements in technology. In this article, we
aim to fill this gap, by conducting a study that seeks to measure public
exposure to deepfakes.

###  1.2 Related Work: Past Measurements of the Spread of Deepfakes

A characteristic feature of the literature on deepfakes is that it largely
focuses on experimental research on the deceptive capacity of deepfakes
(Dobber et al., 2021; Hwang et al., 2021; Shin and Lee, 2022) and the
detection capacity of humans (Mai et al., 2023; Appel and Prietzel, 2022;
Hashmi et al., 2024; Köbis et al., 2021). While such research provides
relevant insights for what could potentially work to address the threats posed
by deepfakes, we know little about people’s actual exposure to and perceptions
of deepfakes outside of controlled research environments.

One of the few primary surveys on public exposure looks at public experiences
with deepfake and digitally altered imagery abuse. Flynn et al (2022) found
that 14.1% (n = 864) of respondents experienced one or more forms of abuse:
non-consensual creation (11.3%), non-consensual distribution (10.4%), and
threats to distribute (10.1%) deepfake and digitally altered images. The
survey, conducted in 2019, was an online survey of 6,109 respondents aged
16–64 years across the UK (n = 2,028), New Zealand (n = 2,027), and Australia
(n = 2,054). The survey found that both females and males experienced
different forms of deepfake image abuse, with males reporting higher overall
rates of victimization and perpetration. However, as mentioned above, deepfake
technology has advanced considerably since 2019, raising the risk that this
evidence is out of date. Besides this survey, there is little primary academic
research on public exposure to different types of deepfakes, including
political deepfakes, deepfakes used for financial frauds, and non-harmful
deepfakes created for educational or entertainment purposes.

It is worth highlighting a considerable amount of grey literature on the
subject. For example, a recent YouGov survey (2024) found that two-thirds
(67%) of respondents were concerned about AI-generated misinformation, and
(75%) regarded digitally altered content, such as deepfakes, as a key
contributor to the same. The survey, which was nationally representative of
the UK, was conducted with over 2000 adults. Similarly, a (2023) cross-country
YouGov survey commissioned by Luminate found that a significant majority of
citizens – over 70% in the UK – were worried about the impact of deepfakes on
upcoming elections. In 2022, a survey of 16,000 people across eight countries
(the US, Canada, Mexico, Germany, Italy, Spain, the UK, and Australia)
revealed that 71% of global respondents were unaware of what a deepfake was,
yet most respondents (57%) were confident in their ability to detect the
difference between an authentic video and a deepfake video (iProov, 2022).
Furthermore, a little over 60% of respondents agreed that deepfakes were
dangerous. Most respondents were concerned that deepfakes could be used for
identity theft, accessing bank accounts, setting up credit cards, and
deceiving people into believing something that is not true (iProov, 2022).
While these surveys provide relevant information about public perceptions,
they adopt a generalised definition of deepfakes, instead of perceptions
around specific types of deepfakes. Moreover, evidence on public exposure to
deepfakes mostly remains restricted to secondary data analytics. For example,
in a study on identity fraud, Sumsub (2023) reported a ten fold increase in
the number of global deepfakes detected between 2022 and 2023. Similarly, Home
Security Heroes (2023), indicated that there were 95,820 deepfake videos in
2023, representing a 550% increase in volume since 2019.

Sensity AI’s report (Ajder et al., 2019) provides case studies of how
deepfakes were used in politics. The report also indicates that 61% of non-
pornographic deepfakes hosted on YouTube target men, including politicians and
corporate figures. Considering non-consensual sexual deepfakes, news reports
indicate that apps like Telegram have been used to generate fake nude images
of more than 680,000 women. Similarly, TikTok creators are being targeted in
non-consensual sexual deepfakes, which is concerning since one-third of
TikTok’s users are children (Kweilin, 2022). The evidence on deepfake creation
is also limited to news and industry reports. Deepfake communities exist on
deepfake pornography websites and popular forums like Reddit, 4chan, 8chan,
and Voat. In 2019, these communities had approximately 100,000 members across
13 platforms (Ajder et al., 2019). However, Home Security Heroes (2023)
estimates a larger ecosystem: with 15 dedicated deepfake creation community
websites and forums, and a membership base exceeding 609,464. Home Security
Heroes reports that one in three deepfake tools enable users to create
deepfake pornography. For example, creating a 60-second deepfake pornographic
video, requires only one clear face image, takes less than 25 minutes and
costs nothing. The report also found that three-fourths of deepfake
pornography users felt no guilt about its usage.

When considering trust in online content, a YouGov (2024) survey suggests that
81% of respondents expressed concern about the trustworthiness of online
content. While this is an important signal, the survey did not specifically
focus on deepfakes, but more broadly on misinformation and AI generated
content. Furthermore, there is little research on public perceptions around
solutions to this issue. To wit, existing research on deepfakes provides
relevant insights on overall trends in the growth of deepfakes, public
awareness of and concerns with deepfakes, and interventions that could help
detect deepfakes. But this research is largely based on industry reports and
experimental research, and does not examine exposure to different types of
deepfakes.

###  1.3 Research aims

By conducting a survey on public exposure to and perceptions of deepfakes,
this article makes a unique contribution to the literature. Three other
features distinguish this article. First, it measures public exposure to
different categories of deepfakes and public concerns about specific
consequences of deepfakes, offering more precise insights for policy. While it
is well known that politicians and celebrities have been targeted by
deepfakes, it is becoming more common for members of the public, particularly
teenagers, to be targeted too. Second, it measures public exposure to specific
political deepfakes circulated in the UK. Third, it is designed to understand
public expectations around actions that social media platforms and other
stakeholders should take to tackle this issue, enabling informed decisions on
regulations and policy.

##  2 Data and Methods

###  2.1 Data collection, ethics, and open science

Data collection was conducted online between 31 May and 5 June 2024 and the
survey was created and administered using Qualtrics111www.qualtrics.com.
Participants were recruited through Prolific222https://www.prolific.com. The
survey was approved by the Ethics Committee at The Alan Turing Institute, UK
(approval number: 24062405). Informed consent was obtained at the start of the
survey according to approved ethical procedures. The materials and data will
be available open access on publication.

###  2.2 Sample

A total of 1403 participants who completed the survey passed standard checks
for data quality and were included in the final sample. The sample was
designed to be nationally representative of the population of the UK across
demographic variables of age, gender and ethnicity (using Prolific’s
representative sample tool). To be eligible to participate in the survey,
participants had to be 18 or over, fluent in English and a resident in the UK.

Respondents were aged between 18 and 83 years old, with a mean age of 46.48
(SD = 15.59). A total of 714 participants identified as female (50.9 %) and
669 as male (47.7 %). 12 participants identified as non-binary/third gender
($<1\%$), with four selecting ‘prefer not to say’ ($<1\%$), and four selecting
‘prefer to self-describe’ ($<1\%$). The majority were White (1193, 85.0%),
while 114 (8.1%) were Asian or Asian British, 47 (3.3%%) were Black, African,
Caribbean, or Black British, and 24 (1.7%) were mixed, multiple or other
ethnicities. 13 ($<1\%$) participants selected ‘any other ethnic group’,
whilst 12 ($<1\%$) chose ‘prefer not to say’. Although participants indicated
more specific ethnic identities, we have combined them into broader categories
to simplify reporting. In terms of education, 781 respondents had degree level
qualifications (55.7%), 215 participants had non-degree level qualifications
(vocational or similar) (15.3%), 403 had no degree level qualifications or
equivalent (including completion of secondary school and below) (28.7%), and 4
participants selected the ‘prefer not to say’ option ($<1\%$). Whilst
participants indicated more specific education levels, we combine them into
broader categories to simplify reporting.

###  2.3 Survey

Demographics and background questions: We collected standard demographic
information about age, gender, ethnicity, education, and political orientation
for each participant. Age could be entered as any number with a minimum of 18.
For gender, ethnicity, and education level, participants were asked to select
the option that they felt best described them from a list of standard
predefined categories. For political orientation, participants were asked to
select the option that best described where they sat on the political spectrum
(more to the left, centre, or more to the right). Participants were also asked
how many hours of their personal time they typically spent using the internet
per day. All demographic questions, other than their age, provided
participants with a ‘prefer not to say’ option (this was not included for age,
as being 18 or over was a requirement to participate).

Awareness: We asked participants whether they had heard the term deepfakes
before the survey (Yes/No/Not sure). We also asked participants who had
previously heard of deepfakes whether they knew what the term meant (Yes/No).

Exposure: To measure exposure to deepfakes, we presented participants with six
different types of deepfakes and asked them to indicate if they had personally
encountered each one (Yes/No/Not sure). The different deepfake types included:
Deepfakes of politicians; Deepfakes of public figures from the entertainment
industry such as actors, social media influencers, musicians; Deepfake
pornographic images/videos (nude/sexually explicit images/videos of someone,
created without their consent using artificial intelligence); Deepfakes
falsifying identities of someone you know for financial frauds/scams (E.g.,
scammers/fraudsters cloning a family member’s voice to trick you into thinking
they are in an emergency situation and need you to send them money); Other
deepfakes which you think have the potential to be harmful (E.g., deepfake
videos of news reporters/scientists spreading health misinformation or
religious leaders spreading propaganda); and Video deepfakes created for
entertainment or educational purposes (E.g., deepfake videos of friends with
celebrity voice-overs or swapping a friend’s face onto a celebrity/movie
character/music video).

We also asked participants about their exposure to specific deepfakes of
politicians that were widely circulated online. The options included: Audio of
Sadiq Khan calling for pro-Palestinian marches; Audio of Sir Lindsay Hoyle
badmouthing his aide; Video of a BBC report on a financial scandal involving
Rishi Sunak; Audio of Sir Keir Starmer swearing at staffers; Image of Rishi
Sunak pouring a sub-standard pint; and Image of Sir Keir Starmer sitting with
late sex offender Jimmy Savile. We also presented participants with placebo
deepfakes—deepfake headlines that we invented and that were never actually
circulated. These included: Video of David Cameron withdrawing support for
Ukraine, Audio of Sir Keir Starmer admitting to previous cannabis use, Image
of Sadiq Khan shaking hands with Jeffrey Epstein, Audio of Liz Truss claiming
to have had an affair with Boris Johnson, and Image of Carla Denyer and other
members of the green party attending a conference by private jet. Using the
difference between average exposure to deepfakes and placebo deepfakes, we
measure true recall of participants to political deepfakes. A similar
methodology was followed by Allcott and Gentzkow (2017) in the context of fake
news during the 2016 US elections.

Fears and Concerns: To find out the fears and concerns people have around
deepfakes, we asked participants about how fearful they felt about personally
becoming a target of harmful deepfakes: Target of deepfake pornography; Target
of deepfake fraud/scam (where your identity is impersonated to trick friends
or family into sending money to a fraudster’s account); Target of any other
deepfake that could be potentially harmful (E.g., someone creating a deepfake
audio/video of you saying something offensive). Response options were: Not at
all fearful /Not very fearful /Somewhat fearful /Very fearful. Participants
were also asked how concerned they felt about the spread of deepfakes on the
internet, and the spread of deepfakes having specific consequences:
Contributing to misogyny and online violence against women and girls, for
example through the creation of deepfake pornography; Facilitating fraud by
impersonating others’ identities (for e.g., audio or visual content pretending
to be one of your family members in need, aimed at tricking victims into
sending money to the fraudster’s account); Manipulating public opinion through
distributing misinformation; Impacting election results or manipulating the
political process; Causing general distrust in information; Supporting the
narratives of extremist or terrorist groups; Adding to online child sexual
abuse material; and Impacting the credibility of evidence during legal
proceedings. Response options for both scales were: Not at all / Not very /
Somewhat / Very.

Creation: To understand access to deepfake creation tools, we asked
respondents whether they had ever created a deepfake, either for
entertainment, educational or any other purposes, by using a tool for swapping
a face with someone else’s in a photo or video or generating/editing audio
using a voice sample. Response options were: Yes / No / Not sure.

Detection and Trust: We asked respondents whether they thought they would be
able to spot if a piece of audio and visual (image and video) content they saw
online was a deepfake (Yes / No / Not sure). To understand participants’
perceptions on trust in online audio and visual content, we asked them about
the extent to which they trusted that general audio and visual (image and
video) content was genuine (i.e., not ‘deepfaked’). Response options were: Do
not trust online audio and visual content at all / Do not trust online audio
and visual content very much / Trust online audio and visual content somewhat
/Trust online audio and visual content very much. Additionally, to understand
participants’ views on reliable detection, we asked them to select which
stakeholders, if any, they would trust to reliably detect deepfakes. The
options were: Mainstream media; UK government; Social media platforms; Other
online providers (e.g., Google); Fact checking organisations; Political
parties; AI-based technologies; Scientific experts; Members of the public;
Others (free text option); No-one (exclusive); Not sure (exclusive).

Solutions: To understand public perceptions about the solutions to address
potentially harmful deepfakes, we asked them two questions. First, what they
thought social media platforms should do to tackle the spread of deepfakes
which have the potential to be harmful online. Response options were: Ban or
suspend users who distribute deepfakes that may be harmful; Make it easier for
people to report deepfakes and request the content to be removed; Make it
difficult for users to find specific deepfakes, for example by preventing them
from appearing through search terms; Make it harder for people to find tools
online which can be used to create harmful deepfakes; Add clear warning labels
or watermarks to deepfakes so that origins of the content are clear; Something
else (free text option); Nothing - platforms should not do anything to tackle
deepfakes (exclusive); None of the above (exclusive); Don’t know (exclusive).

Second, what other actions respondents thought should be taken to address the
spread of harmful deepfakes. Options included: Bans for applications that
allow users to create deepfakes that have the potential to be harmful; Bans
for platforms that host deepfakes that have the potential to be harmful, like
non-consensual deepfake pornography; Increased training for law enforcement
about effectively investigating the creation or distribution of non-consensual
deepfakes; Requirements for platforms to systematically report how many
potentially harmful deepfakes they host and how they are attempting to combat
them; Strict legislation making the creation and distribution of harmful
deepfakes illegal; More education in schools about deepfakes, rules about
creating them, and ways to spot them; Widespread rollouts of media literacy
training courses and public awareness campaigns; More funding for research
aimed at tackling deepfakes; Regulatory bodies to strictly monitor content
violations by BigTech companies and hold them accountable; Public naming and
shaming of platforms that fail to deal with harmful deepfakes; Strict
biometric/remote identification procedures to protect against deepfake fraud;
Something else (free text); Nothing should be done to tackle the spread of
online deepfakes (exclusive); None of the above (exclusive); Don’t know
(exclusive).

###  2.4 Procedure

After participants gave their informed consent to take part in the survey,
they responded to the background and demographic questions. Following this,
participants responded to questions about their familiarity with deepfakes,
before proceeding with questions about fears and concerns. Next, participants
were presented with questions about their exposure to deepfakes and specific
political deepfakes. Post this, participants were asked whether they had
created deepfakes, followed by questions about detecting deepfakes, and
finally, solutions to address deepfakes. At the end of the questions,
participants were given an opportunity to provide feedback in a free text box
before continuing to the debrief, and finally completing the submission and
being redirected to Prolific for payment. The survey was designed to take
approximately 7 minutes to complete, and each participant received £1.05 for
their time.

Figure 1: Self reported exposure to different types of deepfakes. F$<$M
indicates that men are significantly more likely to be exposed to deepfakes
than women.

###  2.5 Data analysis

We present descriptive statistics for public exposure to different types of
deepfakes, public fears and concerns surrounding deepfakes, and public trust
in online content. Additionally, we discuss the top options that respondents
select for stakeholders who can be trusted for reliable detection of
deepfakes, and solutions that could help address the challenge of deepfakes.
As well as describing responses to our key survey questions, we use logistic
regressions to test gender and age differences in exposure to and fears of
being targeted by deepfakes.

##  3 Results

###  3.1 Awareness and creation of deepfakes

Awareness about deepfakes is high: 82.7% participants had previously heard of
the term deepfakes; 12.3% had not previously heard of deepfakes, and 5.1% were
not sure if they had heard of the term before. Amongst the participants who
had previously heard of deepfakes, 91.6% knew what the term meant. While
awareness about deepfakes is high, usage of deepfake creation technologies is
relatively low: only 8.1% respondents indicated that they had created
deepfakes for educational, entertainment or other purposes (though 8.1% of the
population creating deepfakes is of course still a significant amount in
absolute terms).

###  3.2 Self-reported exposure to deepfakes

To understand public exposure to deepfakes, we asked respondents whether they
had personally encountered six different types of deepfakes. The results show
that 50.2% of people have personally encountered deepfakes of public figures
from the entertainment industry such as actors, social media influencers
and/or musicians, while exposure to deepfakes of politicians was at 34.1%.
While nearly half of all respondents (49.3%) had encountered deepfakes created
for educational or entertainment purposes, 15.8% of all respondents were
exposed to deepfakes that could be potentially harmful, including deepfakes
that spread health or religious misinformation or propaganda. Furthermore,
exposure to harmful deepfakes, including non-consensual deepfake pornographic
images/videos (18.8%) and deepfakes that falsify identities for frauds/scams
(9%) were relatively low, though still high enough in absolute terms to
indicate a considerable proportion of the population encountering them.

Analysing self-reported exposure to deepfakes by a cross-section of age and
gender (see Figure1), shows that younger participants aged between 18-25
years, report highest exposure to all categories of deepfakes, except
deepfakes that falsify identities for frauds/scams and other potentially
harmful deepfakes. In these cases, respondents aged between 25-34 years had
the highest exposure: 13.6% and 23.9%, respectively.

Figure 2: Public exposure to high profile political deepfakes in the UK.

Considering gender, men report higher exposure to deepfakes than women (we
note that, as sample sizes are very small, we do not have enough data to
present results about those who did not select either male or female as their
gender, though this would be important follow up research). To further
understand gender and age differences in exposure, we ran a series of logistic
regressions, with age and gender as predictor variables. The analysis reveals
that across all deepfake categories there was a negative and statistically
significant relationship between age and self-reported exposure to deepfakes
(p $<0.001$ in all cases except deepfake frauds and scam). And in all cases
except fraudulent deepfakes, men were significantly more likely to self-report
exposure to deepfakes than women.

Exposure to specific political deepfakes:  As mentioned earlier, governments
across the world are particularly worried about the spread of political
deepfakes. To better understand public exposure to political deepfakes in the
UK, we presented participants with a list of deepfakes and asked them to
recall exposure thereto. In addition to deepfakes, we also introduced
participants to placebo deepfakes – ie. deepfakes that were not actually
circulated, to estimate true recall to political deepfakes. Results show that
on average, 9.1% of respondents reported exposure to deepfakes and 3.5%
reported exposure to placebo deepfakes, as depicted in Figure 2. Using the
difference between average exposure to deepfakes and placebo deepfakes as a
measure of true recall (following the methodology presented in Allcott and
Gentzkow, 2017), we find that on average 5.7% of respondents in the UK have
been exposed to political deepfakes that were circulated online.

###  3.3 Concerns and fears about deepfakes

To understand public concerns about deepfakes, we asked respondents about the
extent to which they were concerned about the spread of deepfakes in general,
and their concerns about eight potential consequences of deepfakes. Overall,
90.4% of the respondents were either very concerned or somewhat concerned
about the spread of deepfakes. Even concerns across the eight different
consequences were high (see Figure 4). The most concerning consequence of
deepfakes was in adding to online child sexual abuse material; 94.3%
respondents indicated being very or somewhat concerned about this issue. This
was followed by concerns about deepfakes increasing distrust in information
(92.1%) and manipulating public opinion (91.8%). Respondents also expressed
concerns about deepfakes increasing misogyny and online violence against women
and girls (89.6%), facilitating fraud (89.4%) and supporting the narratives of
extremist or terrorist groups (89.1%). Concerns about deepfakes impacting
election results were also high (87.4%). In fact, 54.7% people were very
concerned about this issue.

(a)

Figure 3: Public concerns about deepfake consequences

(a)

Figure 4: Public fears about becoming a deepfake target. F$>$M indicates that
women are significantly more likely to be fearful of being targeted by
deepfakes than men.

Fears around deepfakes: We asked respondents the extent to which they feared
being targeted by harmful deepfakes. The results show (see Figure 4) that
people were most fearful about being targeted by deepfake frauds/scams.
Consider how 60.7% respondents were either very or somewhat fearful of being
targeted by such deepfakes. Half (50.1%) of all respondents reported being
somewhat or very much fearful of becoming targets of other potentially harmful
deepfakes, such as deepfakes in which a person is misrepresented as saying
something offensive. And a little over one in four respondents (26.1%)
reported being somewhat or very fearful of becoming targets of deepfake
pornography. To compare gender differences in fears about becoming a target of
each of the 3 harmful deepfake categories, we created a binary outcome for
fear, with ‘Somewhat’ and ‘Very much’ as 1: Fearful, and ‘Not at all’ and ‘Not
very much’ as 0: Not fearful. Across all three categories, women were
significantly more likely to report experiencing fears than men. Women were
127.4% (p$<$0.001) more likely to report being fearful of becoming a target of
deepfake pornography, 77.9% (p$<$0.001) more likely to report being fearful of
becoming a target of deepfake fraud/scam and 47.7% (p$<$0.001) more likely to
report being fearful of becoming a target of other potentially harmful
deepfakes, than men. Table 1 in Supplementary Information shows proportions of
men and women fearing being targeted by each deepfake type.

###  3.4 Trust in content and deepfake detection

To understand perceptions of trust in online audio and visual content, we
asked respondents to indicate the extent to which they trusted that online
content was genuine, ie. not ‘deepfaked’. Results, which were mixed, showed
that 30.1% respondents placed very little trust or no trust at all in the
genuineness of online audiovisual content. However, close to two-thirds of
respondents (65.1%) trusted the genuineness of online audiovisual content
somewhat, while 4.8% of respondents trusted the genuineness of online
audiovisual content very much. We also asked respondents how confident they
felt about spotting deepfakes. While 17.4% felt confident about spotting
deepfakes, 16.5% were not. Most people (66.1%) were unsure if they would be
able to spot deepfakes.

To understand which stakeholders people trusted to reliably detect deepfakes,
we presented them with a list. The top three stakeholders people chose were:
fact checking organisations (54.4%), scientific experts (46.3%) and AI-based
technologies (36.9%). Table 2 in the Supplementary Information shows the
proportion of respondents who place trust in each stakeholder. A few
respondents, who chose ‘others’, indicated that they would trust the
police/MI5, their friends, and other experts including people with prior
knowledge of image manipulation technology, tech-savvy hackers, photo and
video editors, specific Discord or YouTube accounts that specialise in
deepfakes and visual effects, photographic analysis experts and private tech
companies.

We also asked respondents how they would respond if they suspected that a
piece of content was ‘deepfaked’. A majority of respondents indicated that
they would: not engage with the content (54.9%) and report it to the social
media platform they discovered it on (52.1%). The next most popular responses
were researching the content’s origin (31.6%) and using a fact-checking tool
(29.5%). Table 3 in the Supplementary Information shows the proportion of
respondents for each response type. A few respondents who selected ‘other’ in
response to this question reported that they would add a community note on X
(formerly Twitter) or report to Action Fraud UK.

###  3.5 Solutions

To understand what people thought should be done about the problem of
deepfakes, we asked respondents two sets of questions.

The first concerned the actions that social media platforms should take to
address the issue. Overall, participants supported social media platforms
taking action to tackle the harmful content they host; less than 1% chose
‘Nothing - platforms should not do anything to tackle deepfakes’. Most
respondents (87.3%) expressed that platforms should ban or suspend users who
create harmful deepfakes, and 82.4% of participants noted that social media
platforms should make it easier for people to report harmful deepfakes and
request for content removal. Table 4 in the Supplementary Information shows
the overall proportion of participants favouring each action.

A few respondents, who selected ‘something else’, suggested that platforms
should educate people, use tools provided by AI providers to discern synthetic
media and enforce content moderation, while others expressed imposing
bans/fines for platforms that host deepfakes.

Second, we asked participants to indicate other solutions that could help
address the deepfake challenge (see Table 5 in the Supplementary Information).
The top two most commonly chosen solutions were strict legislation making the
creation and distribution of harmful deepfakes illegal (72.1%) and bans for
platforms that host deepfakes that have the potential to be harmful, like non-
consensual deepfake pornography (71.8%). Besides this, 69.3% and 69.1%
respondents chose the following two actions, respectively: more education in
schools about deepfakes, and increased training for law enforcement about
effectively investigating the creation or distribution of non-consensual
deepfakes.

Suggestions of respondents who selected ‘something else’ includes fines for
platforms, a database of known deepfakes, disclaimers on posts featuring
politicians and royals, making it illegal to create deepfakes, and requiring
people to take personal responsibility and be more aware about these issues.

##  4 Discussion and conclusion

To understand public exposure to and perceptions of deepfakes amongst the UK
public, we asked a nationally representative sample of 1403 people whether
they had encountered different categories of deepfakes online, what their
fears and concerns surrounding deepfakes were, and which actions they thought
social media platforms and the government should take to address the spread of
potentially harmful deepfakes online.

Results suggest that most respondents were aware about deepfakes and knew what
the term meant. This stands in stark contrasts to the (2022) iProov survey,
which found that 71% of global respondents, including those from the UK, were
unaware of what a deepfake was. Between 2022 and 2024, high profile deepfakes
of politicians and celebrities such as Taylor Swift, may have brought
deepfakes to the centre of the public discourse, accounting for increased
awareness. Self-reported exposure to deepfakes was high (49.3%) in cases of
non-harmful deepfakes including those created for educational or entertainment
purposes. In terms of targets, exposure to deepfakes featuring public figures
from the entertainment industry (50.2%) exceeded exposure to deepfakes
featuring politicians (34.1%). Exposure to more disconcerting deepfakes,
including non-consensual deepfake pornography, fraud-related deepfakes, and
deepfakes used to perpetuate misinformation or propaganda was relatively low.
In terms of demographics, men and younger participants reported highest
exposure to deepfakes.

Despite relatively low exposure to harmful deepfakes, concerns about deepfakes
were consistently high. The most concerning consequences were that deepfakes
could add to online child sexual abuse material, increase distrust in
information, and manipulate public opinion–with at least 91.8% of respondents
expressing being either ‘very or somewhat’ concerned about each of these
issues. In terms of deepfakes impacting election results, 87.4% respondents
were concerned. When read with a (2023) cross-country survey poll commissioned
by Luminate, which found that over 70% of citizens in the UK were worried
about the impact of deepfakes on upcoming elections – it appears that these
concerns may be growing. However, it is important to note that only 5.7%
respondents recall being exposed to specific political deepfakes circulated
within the UK. The effects of specific political deepfakes on public
perceptions remains to be researched. In 2019, the Responsible Technology
Adoption Unit (previously known as the Centre for Data Ethics and Innovation)
noted that only a few political deepfakes had emerged on social media
platforms, and they were yet to see a “convincing deepfake of a politician
that could distort public discourse.” In 2024, weeks before the elections in
the UK, these observations appear to still hold true.

In terms of public fears, our survey finds that people were most fearful
(60.7%) about being targeted by deepfake frauds/scams, followed by other
potentially harmful deepfakes, such as deepfakes in which a person is
misrepresented as saying something offensive (50.1%) and deepfake pornography
(26.1%). And across all three categories, gender differences in fears revealed
that women were significantly more likely to experience fears than men.
Despite relatively low exposure to these harmful deepfakes, public fears
remain high. Furthermore, close to two-thirds of all respondents were unsure
if they would be able to spot deepfakes.

We also find nuances in people’s perceptions of trust. While most respondents
were concerned about deepfakes causing general distrust in information, close
to 70% report trusting online audiovisual content. What this suggests is
people’s attitudes about trust varies when thought through the lens of the
individual vis-a-vis society. Future research must examine the effects of
exposure to deepfakes on individual and group-level trust in audiovisual
content. It is also important to note here that not all deepfakes are harmful.
Deepfakes may be used for educational or entertainment purposes, as our survey
also reveals. The study did not cover people’s attitudes towards such
beneficial cases, although this may also shape overall trust in audiovisual
content.

In terms of policy-relevant insights, the survey finds that people trust fact
checking organisations (54.4%), scientific experts (46.3%) and AI-based
technologies (36.9%) to reliably detect deepfakes. This aligns with recent
research, which suggests that AI models are better at detecting deepfakes
compared to humans (Hashmi et al., 2024). Considering possible solutions, the
most common actions respondents selected include banning or suspending users
who created harmful deepfakes and requiring platforms to make it easier for
people to report harmful deepfakes and request for content removal. The latter
is a particularly important civil remedy that should be institutionalised,
since not all people targeted by harmful deepfakes would prefer to pursue
criminal proceedings.

Other actions suggested by respondents include strict legislation making the
creation and distribution of harmful deepfakes illegal, bans for platforms
that host deepfakes that have the potential to be harmful such as non-
consensual deepfake pornography, more education in schools about deepfakes and
increased training for law enforcement about effectively investigating the
creation and/or distribution of non-consensual deepfakes. While the needle on
legislation has certainly moved forward, much more needs to be done for
effective enforcement. Busch and Ware (2023) note that governments should
train the public in identifying signs of deepfakes and corroborating content
from different sources. “Pre-bunking” measures may help the public better
respond to deepfakes when they encounter them.

There are several limitations to this research. First, we note that we asked
about exposure to different types of deepfakes. However, there may be overlaps
in the different categories we asked about. For example, there may be cases
where a politician is featured in a non-consensual sexually explicit deepfake.
Because of this, we cannot separate out these intersections between targets
and nature of deepfakes. Second, due to limitations posed by a small sample
size, we were unable to assess whether ethnic minority groups were more
exposed to certain types of deepfakes or experienced greater fear.
Furthermore, we were unable to analyse the experiences of gender minorities
due to limitations of sample size. Future research should examine such
intersections and diverse voices. It must also examine experiences of
teenagers and young adults, who are known to be targets of non-consensual
sexually explicit deepfakes. Third, self-reported exposure may be under- or
over- reported as people may be unsure about their exposure due to
difficulties in identifying deepfakes.

Taken together, our findings show that there is increasing awareness of and
concerns about deepfakes amongst the UK public. At the same time, relatively
high trust in the genuineness of online content shows that people might need
to be more sceptical of claims made in audiovisual content online – and media
literacy programmes could help here. People also expect stronger action from
both platforms and governments to address the challenges posed by harmful
deepfakes.

## Corresponding author:

Tvesha Sippy<EMAIL_ADDRESS>
## Acknowledgements

This work was supported by the Ecosystem Leadership Award under the EPSRC
Grant EPX03870X1 and The Alan Turing Institute.

## References

Ajder, H., Patrini, G., Cavalli, F., & Cullen, L. (2019). The State of
Deepfakes: Landscapes, Threat and Impact. Deeptrace Labs

Allcott, H., & Gentzkow, M. (2017). Social Media and Fake News in the 2016
Election. Journal of Economic Perspectives, 31(2), 211–236.

Appel, M., & Prietzel, F. (2022). The detection of political deepfakes.
Journal of Computer-Mediated Communication, 27(4).

Bateman, J. (2020). Deepfakes and Synthetic Media in the Financial System:
Assessing Threat Scenarios. Carnegie Endowment for International Peace

Brown, J. G., J. N. Bailenson, & J. Hancock. 2023. “Misinformation in Virtual
Reality.” Journal of Online Trust and Safety 1:1–31.

Busch, E., & Ware, J. (2023). The Weaponisation of Deepfakes. Digital
Deception by the Far-Right. Policy Brief. International Centre for Counter-
Terrorism

Dobber, T., Metoui, N., Trilling, D., Helberger, N., & de Vreese, C. (2021).
Do (Microtargeted) Deepfakes Have Real Effects on Political Attitudes? The
International Journal of Press/Politics, 26(1), 69–91.

EndViolenceAgainstWomen. (2024). Government criminalises creation of
deepfakes, but with a major loophole. End Violence Against Women.

Enock, F.E., Bright, J., Stevens, F., Johansson, P., & Margetts, H. Z. (2024).
How do people protect themselves against online misinformation? Attitudes,
experiences and uptake of interventions amongst the UK adult population. The
Alan Turing Institute.

Flynn, A., Powell, A., Scott, A. J., & Cama, E. (2022). Deepfakes and
Digitally Altered Imagery Abuse: A Cross-Country Exploration of an Emerging
form of Image-Based Sexual Abuse. The British Journal of Criminology, 62(6),
1341–1358.

Good Law Project. (2024). Clamping down on political deepfakes.

Hameleers, M., Van Der Meer, T. G. L. A., & Dobber, T. (2022). You Won’t
Believe What They Just Said! The Effects of Political Deepfakes Embedded as
Vox Populi on Social Media. Social Media + Society, 8(3)

Hashmi, A., Shahzad, S. A., Lin, C.-W., Tsao, Y., & Wang, H.-M. (2024).
Unmasking Illusions: Understanding Human Perception of Audiovisual Deepfakes
(arXiv:2405.04097). arXiv.

Home Security Heroes. (2023). State Of Deepfakes: Realities, Threats, And
Impact.

Hwang, Y., Ryu, J.Y., and Jeong, S.-H. (2021). Effects of disinformation using
deepfake: the protective effect of media literacy education. Cyberpsychol.
Behav. Social Netw. 24, 188–193.

iProov. (2022). Deepfake Statistics & Solutions — Protect Against Deepfakes.

Knibbs, K. (2024). Researchers Say the Deepfake Biden Robocall Was Likely Made
With Tools From AI Startup ElevenLabs. WIRED.

Köbis, N. C., Doležalová, B., & Soraperra, I. (2021). Fooled twice: People
cannot detect deepfakes but think they can. iScience, 24(11), 103364.

Kweilin, T. L. (2022). Deepfakes and Domestic Violence: Perpetrating Intimate
Partner Abuse Using Video Technology.

Luminate. (2023). Bots versus ballots: Europeans fear AI threat to elections
and lack of control over personal data.

Mai, K. T., Bray, S., Davies, T., & Griffin, L. D. (2023). Warning: Humans
cannot reliably detect speech deepfakes. PLOS ONE, 18(8), e0285333.

Paris, B., & Donovan, J. (2019). Deep Fakes and Cheap Fakes: The Manipulation
of Audio and Visual Evidence. Data&Society.

Shin, S. Y., & Lee, J. (2022). The Effect of Deepfake Video on News
Credibility and Corrective Influence of Cost-Based Knowledge about Deepfakes.
Digital Journalism, 10(3), 412–432.

Spring, M. (2024). Sadiq Khan says fake AI audio of him nearly led to serious
disorder. BBC News.

Sumsub. (2023). Sumsub Identity Fraud Report 2023.

The Centre for Data Ethics and Innovation. (2019). Independent Report:
Deepfakes and Audiovisual Disinformation. GOV.UK.

Vaccari, C., & Chadwick, A. (2020). Deepfakes and Disinformation: Exploring
the Impact of Synthetic Political Video on Deception, Uncertainty, and Trust
in News. Social Media + Society, 6(1)

Weikmann, T., Greber, H., & Nikolaou, A. (2024). After Deception: How Falling
for a Deepfake Affects the Way We See, Hear, and Experience Media. The
International Journal of Press/Politics, 19401612241233539.

YouGov. (2024). Can you trust your social media feed? UK public concerned
about AI content and misinformation

##  5 Supplementary Information

Deepfake type |  Fear |  Male |  Female
---|---|---|---
Deepfake fraud/scam |  Not fearful |  46.2% |  32.77%
|  Fearful |  53.8% |  67.23%
Deepfake pornography |  Not fearful |  81.8% |  66.81%
|  Fearful |  18.2% |  33.19%
Other potentially harmful deepfake |  Not fearful |  54.7% |  45.24%
|  Fearful |  45.3% |  54.76%
Table 1: Level of fear reported by men and women about becoming a deepfake target. Women are significantly more fearful than men at the p$<$.001 level. Stakeholder |  Response (%)
---|---
Fact checking organisations | 54.4%
Scientific experts | 46.3%
AI-based technologies | 36.9%
UK government | 25.2%
Mainstream media | 22.2%
Other online providers (e.g., Google) | 14.3%
Social media platforms | 14.1%
Members of the public | 13.3%
No-one | 12.6%
Not sure | 10.5%
Political parties | 5.0%
Other, please specify | 1.8%
Table 2: Stakeholders who people trust to reliably detect deepfakes Response to deepfakes |  Response (%)
---|---
Decide not to engage with it (e.g., through liking/sharing it) | 54.9%
Report it to the platform or site you saw it on | 52.1%
Attempt to research its origin (e.g., using a search engine) | 31.6%
Use a fact-checking tool | 29.5%
Ask the opinion of someone else, like a friend or family member | 27.0%
Comment on the content to warn others | 20.7%
Report it to an independent regulator (Ofcom) | 12.1%
Nothing | 9.3%
Report it to the police | 7.8%
Not sure | 4.5%
Something else (please specify) | 1.1%
Table 3: Response to potential deepfakes Platform-based solutions |  Response (%)
---|---
Ban or suspend users who distribute deepfakes that may be harmful | 87.3%
Make it easier for people to report deepfakes and request the content to be removed | 82.4%
Add warning labels or watermarks to deepfakes so that origins of the content are clear | 76.6%
Make it harder for people to find tools which can be used to create harmful deepfakes | 56.8%
Make it difficult for users to find specific deepfakes, for example by preventing them from appearing through search terms | 55.5%
Don’t know | 1.9%
Something else (please specify) | 1.7%
Nothing - platforms should not do anything to tackle deepfakes | 0.3%
None of the above | 0.1%
Table 4: Platform-based solutions for responding to the challenge of deepfakes Other solutions |  Response (%)
---|---
Strict legislation making the creation and distribution of harmful deepfakes illegal | 72.1%
Ban platforms that host deepfakes that have the potential to be harmful, like non-consensual deepfake pornography | 71.8%
More education in schools about deepfakes, rules about creating them, and ways to spot them | 69.3%
Increased training for law enforcement about investigating the creation or distribution of non-consensual deepfakes | 69.1%
Requirements for platforms to report how many potentially harmful deepfakes they host and how they are attempting to combat them | 62.1%
Ban applications that allow users to create deepfakes that have the potential to be harmful | 60.8%
Public naming of platforms that fail to deal with harmful deepfakes | 59.6%
Widespread rollouts of media literacy training courses and public awareness campaigns | 51.1%
More funding for research aimed at tackling deepfakes | 45.0%
Don’t know | 2.4%
Something else (please specify) | 1.1%
None of the above | 0.4%
Nothing should be done to tackle the spread of online deepfakes | 0.3%
Table 5: Other actions for responding to the challenge of deepfakes