# Improved Constants for Effective Irrationality Measures from Hypergeometric
Functions

Paul M. Voutier  London, UK
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we simplify and improve the constant, $c$, that appears in
effective irrationality measures, $|(a/b)^{m/n}-p/q|>c|q|^{-(\kappa+1)}$,
obtained from the hypergeometric method for $a/b$ near $1$. The dependence of
$c$ on $|a|$ in our result is best possible (as is the dependence on $n$ in
many cases). For some applications, the dependence of this constant on $|a|$
becomes important. We also establish some new inequalities for hypergeometric
functions that are useful in other diophantine settings.

###### Key words and phrases:

Diophantine Approximation, Effective Irrationality Measures, Hypergeometric
Functions

###### 2010 Mathematics Subject Classification:

11J82, 11J68, 33C05

##  1\. Introduction

Hypergeometric functions have played an important role in addressing
diophantine problems since the work of Thue. It was Siegel [10] who first
recognised that the functions Thue used were hypergeometric functions. Siegel
also refined Thue’s ideas and used hypergeometric functions himself. For
example, he used them to investigate the integer solutions of Thue equations
involving binomial forms (i.e., $ax^{n}-by^{n}=c$). This work was developed
further by Evertse [7] and others, most notably by Bennett [2].

Baker [1] first showed that hypergeometric functions can also be used to
obtain effective irrationality measures for rational powers of certain
rational numbers. For example, he proved that

$\left|2^{1/3}-\frac{p}{q}\right|>\frac{10^{-6}}{|q|^{2.955}},$

for all integers $p$ and $q$ with $q\neq 0$. Since then, his technique has
been improved, notably through Chudnovsky’s analysis of denominators of the
coefficients of the associated hypergeometric functions [5].

There is also a generalisation of the ordinary hypergeometric method developed
by Baker, known as Thue’s Fundamentaltheorem (from the title of his paper on
it [13]). It can apply to cases not covered by the former.

In previous work [15, 16, 17], we simplified the statement of Thue’s
Fundamentaltheorem and investigated the conditions under which it yields
effective irrationality measures for algebraic numbers. The focus in these
papers was primarily on the irrationality exponent. But for some problems, it
can also be important to have good values for the constant term too ($c$ in
Theorem 2.1 below), in particular a good dependence on the quantity $a$ in
Theorem 2.1. We consider that in this paper.

Furthermore, we obtain some lower bounds for the hypergeometric functions
involved, as well as lower bounds for their denominators (where appropriate).
These have played an important role in some forthcoming works of the author,
so hopefully they will be helpful for other diophantine problems and perhaps
even in other areas too.

##  2\. Results

To present our results, we begin with some notation. For relatively prime
positive integers $m$ and $n$ with $0<m<n/2$ and $n\geq 3$, and a non-negative
integer $r$, we put

$X_{m,n,r}(z)={}_{2}F_{1}(-r,-r-m/n;1-m/n;z)\hskip 8.53581pt\text{and}\hskip
8.53581ptY_{m,n,r}(z)=z^{r}X_{m,n,r}(1/z),$

where ${}_{2}F_{1}$ denotes the classical hypergeometric function. The
condition $m<n/2$, rather than $m<n$, poses no real restriction, and is
necessary for the proof of Lemma 3.5.

Since $-r$ is a non-positive integer,
$X_{m,n,r}(z),Y_{m,n,r}(z)\in\mathbb{Q}[z]$. We let $D_{m,n,r}$ be a positive
integer such that $D_{m,n,r}X_{m,n,r}(z)\in\mathbb{Z}[z]$.

For a non-negative integer $r$ and non-zero $d\in\mathbb{Z}$, we let
$N_{d,m,n,r}$ be a positive integer such that
$\left(D_{m,n,r}/N_{d,m,n,r}\right)X_{m,n,r}\left(1-\sqrt{d}\,z\right)\in\mathbb{Z}\left[\sqrt{\operatorname{sf}(d)}\right][z]$.
Here $\operatorname{sf}(d)$ is the unique squarefree integer such that
$d/\operatorname{sf}(d)$ is a square, with $\operatorname{sf}(1)=1$.

This fixes a notational error in [17] (fixed in arXiv link provided), although
the proofs and the results there are correct and unaffected. This is also a
slight improvement on the definition of what should be denoted as
$N_{m,d,n,r}$ in [17], affecting only the constant in our results. In
practice, when applying our results below in
$\mathbb{Q}\left(\sqrt{t}\right)$, we will take $d$ as a suitable square
multiple of $t$. In this way, the sequence of approximations we obtain in the
course of the proof will be algebraic integers in
$\mathbb{Q}\left(\sqrt{t}\right)$, as required. This explains the choice of
$d$ in Theorem 2.1.

We will use $v_{p}(x)$ to denote the largest power of a prime $p$ which
divides into the rational number $x$. We put

(2.1) ${\mathcal{N}}_{d,n}=\prod_{p|n}p^{\min(v_{p}(d)/2,v_{p}(n)+1/(p-1))},$

and choose real numbers ${\mathcal{C}}_{n}\geq 1$ and ${\mathcal{D}}_{n}>0$
such that

(2.2) $\max_{\stackrel{{\scriptstyle
0<m<n/2}}{{\gcd(m,n)=1}}}\left(\max\left(1,\frac{\Gamma(1-m/n)\,r!}{\Gamma(r+1-m/n)},\frac{n\Gamma(r+1+m/n)}{m\Gamma(m/n)r!}\right)\frac{D_{m,n,r}}{N_{d,m,n,r}}\right)<{\mathcal{C}}_{n}\left(\frac{{\mathcal{D}}_{n}}{{\mathcal{N}}_{d,n}}\right)^{r}$

holds for all non-negative integers $r$, where $\Gamma(x)$ is the Gamma
function. This condition on ${\mathcal{C}}_{n}$ and ${\mathcal{D}}_{n}$ also
corrects the one given in [17].

In what follows, for $z$ not on the negative real line, when we take a root of
$z$, we mean the principal value of the root.

######  Theorem 2.1.

Let $\mathbb{K}$ be an imaginary quadratic field with $m$ and $n$ as above.
Let $a$ and $b$ be algebraic integers in $\mathbb{K}$ with either $0<b/a<1$ a
rational number or $|b/a|=1$ with $0<|b/a-1|<1$. Let ${\mathcal{C}}_{n}$,
${\mathcal{D}}_{n}$ and ${\mathcal{N}}_{d,n}$ be as above with $d=(a-b)^{2}$.
Put

$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{{\mathcal{N}}_{d,n}}{{\mathcal{D}}_{n}}\left\\{\min\left(\left|\sqrt{a}-\sqrt{b}\right|,\left|\sqrt{a}+\sqrt{b}\right|\right)\right\\}^{-2},$
$\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{{\mathcal{D}}_{n}}{{\mathcal{N}}_{d,n}}\left\\{\max\left(\left|\sqrt{a}-\sqrt{b}\right|,\left|\sqrt{a}+\sqrt{b}\right|\right)\right\\}^{2},$
$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle\frac{\log Q}{\log
E}\hskip 8.53581pt\text{ and }$ $\displaystyle c$ $\displaystyle=$
$\displaystyle
3|a|{\mathcal{C}}_{n}\left(20{\mathcal{C}}_{n}\right)^{\kappa}\max\left(n,{\mathcal{N}}_{d,n}^{\kappa}\right).$

If $E>1$, then

$\left|(a/b)^{m/n}-p/q\right|>\frac{1}{c|q|^{\kappa+1}}$

for all algebraic integers $p$ and $q$ in $\mathbb{K}$ with $q\neq 0$.

######  Note.

The dependence on both $a$ and $n$ in $c$ is required. For example, if $n$ is
an odd integer, $a$ a large positive integer and $b=a-1$, then the $0$-th
convergent, $p_{0}/q_{0}$, in the continued fraction expansion of
$(a/b)^{1/n}$ is $1$ and the next partial quotient is $na-(n+3)/2$. So
$\left|(a/b)^{1/n}-p_{0}/q_{0}\right|$ is approximately
$1/\left(na\left|q_{0}\right|^{2}\right)$. Similar results hold for other
small-index convergents too.

An examination of the continued-fraction expansions of such numbers, suggests
that $O(|a|n)$ is the right size for $c$ for any value of $\kappa$ likely to
be obtained in the near-future. We obtain such a value here in cases that
commonly arise in applications like $a-b=1$, $(a-b,n)=1,\cdots$, when
${\mathcal{N}}_{d,n}=1$, so
$c=3{\mathcal{C}}_{n}\left(20{\mathcal{C}}_{n}\right)^{\kappa}|a|n$.

To make our theorem easier to use, we provide values for ${\mathcal{C}}_{n}$
and ${\mathcal{D}}_{n}$. Since it is sometimes useful to have smaller values
of ${\mathcal{C}}_{n}$, we also give ${\mathcal{D}}_{2,n}$, the smallest value
of ${\mathcal{D}}_{n}$ that allows us to take ${\mathcal{C}}_{n}=100$.

It is known (see Theorem 4.3 in [5]) that $D_{m,n,r}$ has the following
asymptotics. As $r\rightarrow\infty$,

$\overline{\lim_{r\rightarrow\infty}}\frac{\log
D_{m,n,r}}{r}\leq(\mathrm{Chr})_{n}^{2},$

where

$(\mathrm{Chr})_{n}^{2}=\frac{\pi}{\varphi(n)}\sum_{\begin{subarray}{c}j=1\\\
(j,n)=1\end{subarray}}^{\lfloor n/2\rfloor}\cot(\pi j/n).$

######  Theorem 2.2.

(a) If $3\leq n\leq 100$, then we can take

$\left({\mathcal{C}}_{n},{\mathcal{D}}_{n}\right)=\left({\mathcal{C}}_{1,n},{\mathcal{D}}_{1,n}\right)\text{
or }\left(100,{\mathcal{D}}_{2,n}\right),$

where ${\mathcal{C}}_{1,n}$, ${\mathcal{D}}_{1,n}$ and ${\mathcal{D}}_{2,n}$
are in Tables 1–3.

(b) If $101\leq n\leq 757$ is prime and we consider only $m=1$ in (2.2), then
we can take

$\left({\mathcal{C}}_{n},{\mathcal{D}}_{n}\right)=\left({\mathcal{C}}_{1,n},{\mathcal{D}}_{1,n}\right)\text{
or }\left(100,{\mathcal{D}}_{2,n}\right),$

where ${\mathcal{C}}_{1,n}$, ${\mathcal{D}}_{1,n}$ and ${\mathcal{D}}_{2,n}$
are in Tables 4–6.

(c) Otherwise, let $d_{1}=\gcd\left(d,n^{2}\right)$ and
$d_{2}=\gcd\left(d/d_{1},n^{2}\right)$. If $d_{2}=1$, then

$\left({\mathcal{C}}_{n},{\mathcal{D}}_{n}\right)=\left(n,n\mu_{n}\right),$

where $\mu_{n}=\displaystyle\prod_{\stackrel{{\scriptstyle
p|n}}{{p\,\,\mathrm{prime}}}}p^{1/(p-1)}$.

Since there are $\varphi(n)/2$ values of $m$ to consider for each value of
$n$, the work required to continue part (a) for larger values of $n$ soon
becomes prohibitive. It is for this reason that we restrict to considering
only $m=1$ for $101\leq n\leq 757$, and also only consider $n$, prime, in this
interval. Certainly for $n$ this large, prime values of $n$ are the most
important ones.

We stop at $n=757$ as the analytic bounds we use fail to improve on
${\mathcal{D}}_{n}=n\mu_{n}$ for the next prime $n=761$ (here our value for
$\log{\mathcal{D}}_{n}$ is $6.651$, while $\log n\mu_{n}=6.644$) and various
larger values of $n$. In theory, using Lemma 3.6, one could extend part (b) to
$n<1289$, as well as obtaining smaller values of ${\mathcal{D}}_{n}$ in parts
(a) and (b), but this would require much more computation.

Before turning to the proof of these theorems, we also mention here some
results obtained in the course of the proof that may be of use to other
researchers.
$\bullet$ Lemma 3.1 improves on previous versions of this “forklore lemma”
that is crucial for obtaining effective irrationality measures from a sequence
of good approximations. Here we use efficiently the $0$-th element in the
sequence of good approximations to replace the usual lower bound on $|q|$ with
a (typically weak) condition on $\ell_{0}$ and $E$.
$\bullet$ Lemma 3.3 provides a new lower bound for the hypergeometric
functions arising in the upper bound for the quality of our sequence of good
approximations. Moreover, it is best-possible where the hypergeometric method
is applicable.
$\bullet$ Lemma 3.5 provides a new lower bound for the hypergeometric
functions used in the construction of our sequence of good approximations.
$\bullet$ Lemma 3.7 provides a lower bound for the denominators of the
hypergeometric functions used in the construction of our sequence of good
approximations. It has the correct dependence on $n$.

##  3\. Preliminary Results

The following lemma is used to obtain an effective approximation measure for a
complex number $\theta$ from a sequence of good approximations in an imaginary
quadratic field.

######  Lemma 3.1.

Let $\theta\in\mathbb{C}$ and let $\mathbb{K}$ be an imaginary quadratic
field. Suppose that for all non-negative integers $r$, there are algebraic
integers $p_{r}$ and $q_{r}$ in $\mathbb{K}$ satisfying $p_{r}q_{r+1}\neq
p_{r+1}q_{r}$ with $\left|q_{r}\right|<k_{0}Q^{r}$ and $\left|q_{r}\theta-
p_{r}\right|\leq\ell_{0}E^{-r}$, for some real numbers $k_{0},\ell_{0}>0$ and
$E,Q>1$ with $2\ell_{0}E\geq 1$. Then for any algebraic integers $p$ and $q$
in $\mathbb{K}$ with $q\neq 0$ and $p/q\neq p_{r}/q_{r}$ for all non-negative
integers $r$, we have

$\left|\theta-\frac{p}{q}\right|>\frac{1}{c|q|^{\kappa+1}},\text{ where
}c=2k_{0}\left(2\ell_{0}E\right)^{\kappa}\text{ and }\kappa=\frac{\log Q}{\log
E}.$

######  Note.

This is Lemma 6.1 in [15] with two changes. It has an improved value of $c$
due to the additional condition that $p/q\neq p_{r}/q_{r}$ for all non-
negative integers $r$. We have also replaced the lower bound on $|q|$ with a
lower bound for $2\ell_{0}E$.

If we remove the restriction that $p/q\neq p_{r}/q_{r}$ for all non-negative
integers $r$, then the lemma still holds, but with $c$ above replaced by
$c=2k_{0}Q\left(2\ell_{0}E\right)^{\kappa}$.

###### Proof.

The proof is quite similar to that of Lemma 6.1 in [15].

Let $p$, $q$ be algebraic integers in $\mathbb{K}$. If $|q|\geq
1/\left(2\ell_{0}\right)$, put $\displaystyle
r_{0}=\left\lfloor\frac{\log(2\ell_{0}|q|)}{\log E}\right\rfloor+1$. If
$|q|<\left(2\ell_{0}\right)$, then put $r_{0}=0$. Note that in the first case,
since $E>1$ and $2\ell_{0}|q|\geq 1$, we have $r_{0}\geq 1$.

In the first case, it follows that
$0\leq\log\left(2\ell_{0}|q|\right)/\log(E)<r_{0}$. Hence, for all $r\geq
r_{0}$,

$\ell_{0}E^{-r}<\ell_{0}E^{-(\log(2\ell_{0}|q|))/(\log E)}=1/(2|q|)<1,$

since $E>1$.

When $r_{0}=0$, then for all $r\geq r_{0}$,

$\ell_{0}E^{-r}\leq\ell_{0}<1/(2|q|)<1,$

since $E>1$ and every non-zero algebraic integer in $\mathbb{K}$ has absolute
value at least $1$.

In both cases, we have

(3.1) $\ell_{0}E^{-r}<1/(2|q|)<1,$

for all $r\geq r_{0}$.

If we have $q_{r}=0$ for some $r\geq r_{0}$, then from (3.1),
$\left|p_{r}\right|=\left|q_{r}\theta-p_{r}\right|<\ell_{0}E^{-r}<1$, which
implies that $p_{r}=0$ (again, using the fact that all non-zero algebraic
integers in these fields are of absolute value at least $1$). This contradicts
the supposition that $p_{r}q_{r+1}\neq p_{r+1}q_{r}$. Therefore, $q_{r}\neq 0$
for all $r\geq r_{0}$.

So, for any $r\geq r_{0}$ with $p/q\neq p_{r}/q_{r}$, we have

(3.2)
$\left|\theta-\frac{p}{q}\right|\geq\left|\frac{p_{r}}{q_{r}}-\frac{p}{q}\right|-\left|\theta-\frac{p_{r}}{q_{r}}\right|\geq\frac{1}{|qq_{r}|}-\frac{\ell_{0}}{E^{r}|q_{r}|}>\frac{1}{2|qq_{r}|},$

again using (3.1) and the fact that $p_{r}q-q_{r}p$ is a non-zero algebraic
integer and hence of absolute value at least $1$ in such fields.

If $|q|\geq 1/\left(2\ell_{0}\right)$, then the choice of $r_{0}$ yields

(3.3)
$Q^{r_{0}}\leq\exp\left(\frac{\log(2\ell_{0}|q|)+\log(E)}{\log(E)}\log(Q)\right)=\left(2E\ell_{0}|q|\right)^{\kappa}.$

If $|q|<1/\left(2\ell_{0}\right)$, so that $r_{0}=0$, then the same upper
bound holds for $Q^{r_{0}}=1$ by our assumption that $2E\ell_{0}\geq 1$ and
hence that $2E\ell_{0}|q|\geq 1$, since $q\neq 0$ implies that $|q|\geq 1$.

Combining (3.2) and (3.3) with our upper bound in the lemma for
$\left|q_{r_{0}}\right|$, we have

$\left|\theta-\frac{p}{q}\right|>\frac{1}{2|qq_{r_{0}}|}\geq\frac{1}{2|q|k_{0}Q^{r_{0}}}\geq\frac{1}{2k_{0}(2E\ell_{0})^{\kappa}|q|^{\kappa+1}},$

when $p/q\neq p_{r_{0}}/q_{r_{0}}$. ∎

For any non-negative integer, $r$, let

(3.4)
$R_{m,n,r}(z)=\frac{(m/n)\cdots(r+m/n)}{(r+1)\cdots(2r+1)}{}_{2}F_{1}\left(r+1-m/n,r+1;2r+2;1-z\right).$

The next lemma contains the relationship that allows the hypergeometric method
to provide good sequences of rational approximations.

######  Lemma 3.2.

Let $m,n$ and $r$ be non-negative integers with $0<m<n$ and $\gcd(m,n)=1$. If
$z$ is any complex number with $|z|\leq 1$ and $|z-1|<1$, then

(3.5) $Y_{m,n,r}(z)-(1/z)^{m/n}X_{m,n,r}(z)=z^{-m/n}(z-1)^{2r+1}R_{m,n,r}(z).$

###### Proof.

This is a slight variation on equation (4.2) in [5] with $\nu=m/n$. We
multiply that equation by $(1/z)^{m/n}$ to obtain (3.5).

The reason for this change is that we have an easy upper bound for
$\left|X_{m,n,r}(z)\right|$ when $0<b/a<1$ is a real number, so we will use
$X_{m,n,r}(z)$ to define our $q_{r}$ in Lemma 3.1. ∎

######  Lemma 3.3.

Let $r$ be a non-negative integer and $0<\nu<1$. For $|z|=1$ and
$\operatorname{Re}(z)\geq 0$, we have

(3.6) $\left|{}_{2}F_{1}\left(r+1-\nu,r+1;2r+2;1-z\right)\right|\geq 1,$

with the minimum value occurring at $z=1$.

######  Remark 1.

In fact, (3.6) is true for all $z\in\mathbb{C}$ with $|z|\leq 1$ and all
${}_{2}F_{1}\left(a,b;2a;1-z\right)$ with $a,b,c\in\mathbb{R}$, $a,b>0$ and
$c\geq\max(a,b)$. But the proof is longer and more complicated, so we have not
included it here.

###### Proof.

Using Pochammer’s integral (see equation (1.6.6) of [11]), along with the
transformation $t=1/s$, we can write

$\displaystyle{}_{2}F_{1}\left(a,b;c;z\right)$
$\displaystyle=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{1}^{\infty}(s-1)^{c-b-1}s^{a-c}(s-z)^{-a}ds$
$\displaystyle=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{0}^{\infty}s^{c-b-1}(s+1)^{a-c}(s+1-z)^{-a}ds.$

Therefore, we can write

${}_{2}F_{1}\left(a,b;c;1-z\right)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{0}^{\infty}s^{c-b}(s+1)^{a-c}(s+z)^{-a}ds/s$

and our problem becomes one of showing that the absolute value of the function

$\int_{0}^{\infty}t^{\alpha}(t+1)^{-\beta}(t+z)^{-\gamma}\frac{dt}{t}$

with $\alpha,\beta,\gamma>0$ and $\beta+\gamma>\alpha$ attains its minimum in
$\Omega=\\{z:|z|\leq 1,z\notin[-1,0]\\}$ at $z=1$.

Note that here we have $\alpha=c-b$, $\beta=c-a$ and $\gamma=a$.

We can change the integration path to any path that avoids the singularities
of the integrand, i.e., any path that stays in the open angle bounded by the
rays $\\{-\tau z:\tau>0\\}$ and $\\{-\tau:\tau>0\\}$ containing the positive
semi-axis. So we will change it to the ray $\\{\tau\sqrt{z}:\tau>0\\}$. Thus
out integral becomes

$\int_{0}^{\infty}\left(\sqrt{z}t\right)^{\alpha}\left(\sqrt{z}t+1\right)^{-\beta}\left(\sqrt{z}t+z\right)^{-\gamma}\frac{dt}{t}=z^{(\alpha-\beta-\gamma)/2}\int_{0}^{\infty}t^{\alpha}\left(t+1/\sqrt{z}\right)^{-\beta}\left(t+z/\sqrt{z}\right)^{-\gamma}\frac{dt}{t}.$

Putting $w=1/\sqrt{z}$ and recalling that $|z|=1$, we have

$\left|z^{(\alpha-\beta-\gamma)/2}\int_{0}^{\infty}t^{\alpha}\left(t+1/\sqrt{z}\right)^{-\beta}\left(t+z/\sqrt{z}\right)^{-\gamma}\frac{dt}{t}\right|=\left|\int_{0}^{\infty}t^{\alpha}\left(t+w\right)^{-\beta}\left(t+wz\right)^{-\gamma}\frac{dt}{t}\right|,$

so the problem is reduced to establishing the following:
let $w,z^{\prime}\in\mathbb{C}$,
$\operatorname{Re}(w),\operatorname{Re}(z^{\prime})>0$ and
$z^{\prime}w\in\mathbb{R}_{+}$. Then

$\left|\int_{0}^{\infty}t^{\alpha}(t+w)^{-\beta}(t+z^{\prime})^{-\gamma}\frac{dt}{t}\right|\geq\left|\int_{0}^{\infty}t^{\alpha}(t+|w|)^{-\beta}(t+|z^{\prime}|)^{-\gamma}\frac{dt}{t}\right|\,.$

Since $c=2a$, $|z|=1$ and our definition of $w$, here we have $\beta=\gamma$
and $z^{\prime}=z/\sqrt{z}=\sqrt{\bar{z}}$, this is immediate because the
integrand on the left is then positive and obviously greater than the one on
the right. ∎

######  Lemma 3.4.

Let $m,n$ and $r$ be non-negative integers with $0<m<n$ and $\gcd(m,n)=1$.

(a) If $|z|=1$ and $|z-1|<1$, then

$\left|X_{m,n,r}(z)\right|<1.072\frac{r!\Gamma(1-m/n)}{\Gamma(r+1-m/n)}\left|1+\sqrt{z}\right|^{2r}.$

(b) If $z\in\mathbb{R}$ with $0\leq z\leq 1$, then

$\left|X_{m,n,r}(z)\right|<\left|1+\sqrt{z}\right|^{2r}.$

###### Proof.

(a) This is a slight refinement of Lemma 7.3(a) of [15]. In the proof of that
lemma, we showed that in our notation here

$\left|X_{m,n,r}(z)\right|\leq\frac{4}{\left|1+\sqrt{z}\right|^{2}}\frac{\Gamma(1-m/n)\,r!}{\Gamma(r+1-m/n)}\left|1+\sqrt{z}\right|^{2r}.$

Since $z$ is on the unit circle, we can write
$1+\sqrt{z}=1+z_{1}\pm\sqrt{1-z_{1}^{2}}i$, where $0\leq z_{1}\leq 1$. Here we
have $|\theta|<\pi/3$ in order that $|z-1|<1$ holds. Hence
$z_{1}=\cos(\theta/2)>\cos(\pi/6)$, and so

$\frac{4}{\left|1+\sqrt{z}\right|^{2}}<1.072.$

(b) This is Lemma 5.2 of [14], noticing that $Y_{m,n,r}(z)$ there is our
$X_{m,n,r}(z)$. ∎

In order to obtain the simplified constant in our effective irrationality
measure, we will also need lower bounds for the hypergeometric functions and
for their denominators. We now establish these results.

######  Lemma 3.5.

Let $m,n$ and $r$ be non-negative integers with $0<m<n/2$ and $\gcd(m,n)=1$.
For $z\in\mathbb{C}$ with $\operatorname{Re}(z)\geq 0$, we have

$\left(1+\operatorname{Re}(z)\right)^{r}\leq\left|X_{m,n,r}(z)\right|$

and

$\left(1+\operatorname{Re}(z)\right)^{r}\leq\left|Y_{m,n,r}(z)\right|.$

###### Proof.

We start by showing that all the zeroes of $X_{m,n,r}(z)$ are negative real
numbers. Equation (4.21.2) in [12, p. 62] defines the Jacobi polynomials,
$P_{n}^{(\alpha,\beta)}$ by

$P_{n}^{(\alpha,\beta)}(1-2z)=\binom{n+\alpha}{n}{}_{2}F_{1}\left(-n,n+\alpha+\beta+1;\alpha+1;z\right).$

Here Szego’s $n$, $\alpha$ and $\beta$ correspond to our $r$, $-m/n$ and
$-2r-1$, respectively.

The zeroes of $X_{m,n,r}(z)$ will all be negative if the zeroes of
$P_{n}^{(\alpha,\beta)}(z)$ are all real and larger than $1$ for this choice
of $n$, $\alpha$ and $\beta$. The number of such zeroes is the quantity
$N_{3}$ in Theorem 6.72 in [12, p.145]. In the notation of this theorem,
$Z=\left[r+1/2+m/n\right]=r$ – this is one of the reasons why we need the
conition $m<n/2$. Furthermore, we see that Szego’s ${2n+\alpha+\beta\choose
n}{n+\alpha\choose n}$ is equal to ${-1-m/n\choose n}{r-m/n\choose r}$ for our
choice of $n$, $\alpha$ and $\beta$. This quantity is negative if $r$ is odd
and positive if $r$ is even. Thus, by equation (6.72.8) of [12, p.146],
$N_{3}=2\left[(r+1)/2\right]=r$, if $r$ is even and
$N_{3}=2\left[r/2\right]+1=r$ if $n$ is odd.

Since all the zeroes of $X_{m,n,r}$ are real, $z$ is always at least as far
from each of these zeroes as $\operatorname{Re}(z)$ is. Therefore,
$\left|X_{m,n,r}(z)\right|\geq\left|X_{m,n,r}(\operatorname{Re}(z))\right|$.
From Lemma 5.2 of [14], we have
$\left|X_{m,n,r}(\operatorname{Re}(z))\right|\geq(1+\operatorname{Re}(z))^{r}$,
as stated (note that $Y_{m,n,r}(z)$ in [14] is the same as $X_{m,n,r}(z)$
here.

Since $Y_{m,n,r}(z)=z^{r}X_{m,n,r}\left(z^{-1}\right)$, all its zeroes are
also negative real numbers, so we have
$\left|Y_{m,n,r}(z)\right|\geq\left|Y_{m,n,r}(\operatorname{Re}(z))\right|$
too. Since the coefficient of $z^{k}$ in $(1+z)^{r}$ equals the coefficient of
$z^{r-k}$, the argument in Lemma 5.2 of [14] showing that
$\left|X_{m,n,r}(\operatorname{Re}(z))\right|\geq(1+\operatorname{Re}(z))^{r}$
also shows that
$\left|Y_{m,n,r}(\operatorname{Re}(z))\right|\geq(1+\operatorname{Re}(z))^{r}$.
∎

Important for our work will be the following result of [3].

######  Lemma 3.6.

Suppose $m$ and $n$ are relatively prime rational integers with $3\leq n\leq
10^{4}$ and $0<m<n$. Recall that
$\displaystyle\theta(x;n,m)=\sum_{\stackrel{{\scriptstyle p\leq x}}{{p\equiv
m\bmod n}}}\log(p)$, where the sum is over all such primes $p$.

We have

(3.7)
$\left|\theta(x;n,m)-\frac{x}{\varphi(n)}\right|<\frac{x}{840\log(x)}<\left\\{\begin{array}[]{ll}4.31\cdot
10^{-5}x&\text{for $x\geq 10^{12}$},\\\ 3.98\cdot 10^{-5}x&\text{for $x\geq
10^{13}$}.\end{array}\right.$

Furthermore,

(3.8) $|\theta(x;n,m)-x/\varphi(n)|<1.818\sqrt{x},$

for $x\leq 10^{12}$ and each $3\leq n\leq 10^{4}$ (for $x\leq 10^{13}$ when
$3\leq n\leq 100$).

###### Proof.

Equation (3.7) follows from Theorem 1.2 of [3].

Equation (3.8) follows from Theorem 1.9 and equation (A.2) of [3]. ∎

######  Lemma 3.7.

Let $m$, $n$ and $r$ be non-negative integers with $0<m<n/2$, $\gcd(m,n)=1$
and $n\geq 3$.

(a) We have

(3.9) $D_{m,n,r}>(n/4)^{r}\cdot\prod_{\stackrel{{\scriptstyle
p|n}}{{\text{$p$,
prime}}}}p^{v_{p}((2r)!)-v_{p}(r!)}\geq\left(n\mu_{n}/4\right)^{r}(2r+1)^{-\omega(n^{\prime})/2},$

where $\mu_{n}$ is as in Theorem 2.2, $n^{\prime}$ is the largest odd factor
of $n$ and $\omega(n^{\prime})$ is the number of distinct prime factors of
$n^{\prime}$.

(b) We have

(3.10) $D_{m,n,r}>\left\\{\begin{array}[]{ll}0.08\cdot 2.1^{r}&\text{if
$n=3$},\\\ 0.02\cdot 3.77^{r}&\text{if $n=4$},\\\ 0.3\cdot 2.54^{r}&\text{if
$n=5$},\\\ 0.3\cdot 10.9^{r}&\text{if $n=6$},\\\ 0.7\cdot 2.63^{r}&\text{if
$n=7$},\\\ 0.2\cdot 5.53^{r}&\text{if $n=8$}.\end{array}\right.$

######  Note.

The lower bound in (3.9), while smaller than the asymptotics of $D_{m,n,r}$,
does show the right dependence on $n$. In Remark 7.7 of [15], we stated that
$n\mu_{n}$ is approximately
$\left(\pi/e^{\gamma}\right)(\mathrm{Chr})_{n}^{2}$, so $n\mu_{n}/4$ is
approximately $0.44(\mathrm{Chr})_{n}^{2}$.

###### Proof.

(a) Our proof uses the fact that if $f(z)\in\mathbb{Q}[z]$, then the least
common multiple of the denominators of its coefficients must be at least the
reciprocal of the absolute value of $f(v)$ for an integer $v$. Since
${}_{2}F_{1}(a,b;c;1)$ has a nice value, we consider $v=1$ here.

Using the Chu-Vandermonde identity (see equation (15.5.24) of [6]) with
$b=-r-m/n$, $c=1-m/n$ and $n$ there equal to our $r$, we have

$X_{m,n,r}(1)=\frac{(c-b)_{r}}{(c)_{r}}=\frac{(r+1)\cdots(2r)}{(1-m/n)\cdots(r-m/n)}=n^{r}\frac{(r+1)\cdots(2r)}{(n-m)\cdots(rn-m)},$

where $(a)_{r}=a\cdots(a+r-1)$ is Pochhammer’s symbol. Since $n$ is relatively
prime to the denominators of the coefficients of $X_{m,n,r}(z)$ (all the terms
in the denominators are of the form $in-m$ and $\gcd(m,n)=1$), we need only
consider $(r+1)\cdots(2r)/\left[(n-m)\cdots(rn-m)\right]$.

Now

$\frac{(n-m)\cdots(nr-m)}{(r+1)\cdots(2r)}>\frac{(n/2)\cdots(nr-n/2)}{(r+1)\cdots(2r)}=\frac{n^{r}(2r-1)!}{2^{2r-1}(r-1)!}\frac{r!}{(2r)!}=(n/4)^{r},$

since $m<n/2$.

But we can remove more powers of prime divisors of $n$ from $(r+1)\cdots(2r)$
too:

$\prod_{\stackrel{{\scriptstyle p|n}}{{\text{$p$,
prime}}}}p^{v_{p}((2r)!)-v_{p}(r!)}|\left((r+1)\cdots(2r)\right).$

Observe that $(2r)!/r!=2^{r}\cdot 1\cdot 3\cdots(2r-1)$, so
$2^{v_{2}((2r)!)-v_{2}(r!)}=2^{r}$. Letting $s_{p}(r)$ be the sum of the
digits in the base $p$ expansion of $r$, we find that
$v_{p}((2r)!)-v_{p}((r!))=(2r-s_{p}(2r))/(p-1)-(r-s_{p}(r))/(p-1)$ (see
Exercise 14 on page 7 of [8]). So

$v_{p}((2r)!)-v_{p}((r!))=r/(p-1)+\left(s_{p}(r)-s_{p}(2r)\right)/(p-1),$

for primes $p>2$. The maximum of $p^{(s_{p}(2r)-s_{p}(r))/(p-1)}$ occurs when
all the base $p$ digits of $2r$ are equal to $p-1$, in which case it is
$\sqrt{2r+1}$. Part (a) of the lemma follows.

(b) Taking $A=0$ and $\ell=1$ in Lemma 3.3(b) of [14], we know that if
$p>(nr+m)^{1/2}$ is a prime such that $p\equiv-m\bmod n$ and

$\frac{nr+m+n}{n-1}\leq p\leq nr-m,$

then $p|D_{m,n,r}$. Furthermore, if $r\geq n$, then
$(nr+m+n)/(n-1)>(nr+m)^{1/2}$. This is why we calculate the values for $r\leq
n$.

Thus

(3.11) $\displaystyle\log D_{m,n,r}$
$\displaystyle\geq\sum_{\stackrel{{\scriptstyle(nr+m+n)/(n-1)\leq p\leq
nr-m}}{{p\equiv-m\bmod n}}}\log(p)$
$\displaystyle=\theta\left(nr-m;n,-m\right)-\theta\left((nr+m+n)/(n-1);n,-m\right),$

where $\theta(x;n,-m)$ is the sum of the logarithms of all primes $p\leq x$
with $p\equiv-m\bmod n$ and $\varphi(n)$ is Euler’s phi function.

From Corollary 1.7 of [3], we have

$\left|\theta(x;n,-m)-\frac{x}{\varphi(n)}\right|<0.00174x,$

for $x\geq 10^{6}$ and the pairs $(m,n)$ being considered here.

We apply this inequality to (3.11) to obtain the bounds in the lemma for
$r\geq 10^{6}$. In fact, we obtain inequalities where the constants in front
of the exponential terms are slightly larger.

Using a program written in Java, we compute the denominators for the remaining
polynomials. This took just under 500 seconds on a Windows laptop with an
Intel i7-9750H 2.60GHz CPU. It is from this calculation that the constants in
front of the exponential terms arise. E.g., for $n=3$, we had to replace the
constant $0.1$ with $0.08$, which is required for $m=1$ and $r=13$. Part (b)
follows. ∎

The following lemma is much weaker than Lemma 3.7 permits (roughly
$\left(n\mu_{n}/8\right)^{2r}$), but it suffices for our needs here.

######  Lemma 3.8.

Let $m,n$ and $r$ be non-negative integers with $0<m<n/2$, $n\geq 3$ and
$\gcd(m,n)=1$. We have

(3.12)
$\frac{m}{60n}<D_{m,n,r}^{2}\frac{(m/n)\cdots(r+m/n)}{(r+1)\cdots(2r+1)}$

######  Note.

We do not consider $r=0$ here, as the right-hand side is $m/n$ in this case
and hence dependent on $n$, whereas we want an absolute constant on the left-
hand side in our result.

###### Proof.

We first consider $r=0$. Here $X_{m,n,r}(z)=1$, so $D_{m,n,r}=1$ and the
right-hand side of (3.12) is $m/n$. So the lemma holds in this case.

Now consider $r=1$. Here $X_{m,n,r}(z)=(n+m)z/(n-m)+1$, so $D_{m,n,r}=(n-m)/2$
if $m$ and $n$ are both odd and $n-m$ otherwise, since $m$ and $n$ are
relatively prime. So right-hand side of (3.12) is at least
$m(n+m)(n-m)^{2}/\left(24n^{2}\right)$. Taking the derivative of this quantity
with respect to $m$, we obtain

$\frac{(n-m)\left(n^{2}-nm-4m^{2}\right)}{24n^{2}}.$

Its numerator is zero when $m=n$ and $m=\left(-1\pm\sqrt{17}\right)n/8$. Only
$m=\left(-1+\sqrt{17}\right)n/8$ satisfies $0<m<n/2$, so we consider this
value of $m$ (where $m(n+m)(n-m)^{2}/\left(24n^{2}\right)=0.0084\ldots
n^{2}>1/14$ for $n\geq 3$), along with $m=1$ (where
$m(n+m)(n-m)^{2}/\left(24n^{2}\right)=(n+1)(n-1)/\left(24n^{2}\right)\geq
2/27$ for $n\geq 3$) and $m=(n-1)/2$ (where
$m(n+m)(n-m)^{2}/\left(24n^{2}\right)=(n-1)(3n-1)(n+1)^{2}/\left(384n^{2}\right)\geq
2/27$ for $n\geq 3$). So the lemma holds for $r=1$.

We need a lower bound for $(m/n)\cdots(r+m/n)/\left((r+1)\cdots(2r+1)\right)$.
We can write

$\frac{(m/n)\cdots(r+m/n)}{(r+1)\cdots(2r+1)}=\frac{\Gamma(r+1+m/n)\Gamma(r+1)}{\Gamma(m/n)\Gamma(2r+2)}.$

We will use

$1<(2\pi)^{-1/2}x^{(1/2)-x}e^{x}\Gamma\left(x\right)<e^{1/(12x)},$

(see inequality (5.6.1) in [6]).

Applying these inequalities to each of these four gamma function values, we
obtain

$\frac{(m/n)\cdots(r+m/n)}{(r+1)\cdots(2r+1)}>\left(\frac{r+1+m/n}{r+1}\right)^{r+1}(r+1+m/n)^{m/n-1/2}\frac{2^{-2r-3/2}}{(m/n)^{m/n-1/2}e^{(n/m+1/(2r+2))/12}}.$

Simplifying this, we find that

(3.13)
$\frac{(m/n)\cdots(r+m/n)}{(r+1)\cdots(2r+1)}>4^{-r}\sqrt{m/(8n)}r^{m/n-1/2}>4^{-r}\sqrt{m/(8nr)}.$

We combine this with the lower bound, $D_{m,n,r}>(n/4)^{r}$, which follows
from (3.9) in Lemma 3.7(a):

$(n/8)^{2r}\sqrt{m/(8rn)}<D_{m,n,r}^{2}\frac{(m/n)\cdots(r+m/n)}{(r+1)\cdots(2r+1)}.$

For $n\geq 9$, the left-hand side is greater than $m/(60n)$ for $r\geq 2$.

For $n\leq 8$, we use Lemma 3.7(b). Writing the lower bounds there as
$d_{1}\cdot d_{2}^{r}<D_{m,n,r}$ and using (3.13), the right-hand side of
(3.12) is greater than

$\sqrt{m/(8rn)}d_{1}\left(d_{2}/2\right)^{2r}.$

For $3\leq n\leq 8$, we can easily calculate that this quantity is greater
than $m/(60n)$ for $n=3$ with $r\geq 20$; $n=4$ with $r\geq 4$; and
$n=5,6,7,8$ with $r\geq 2$. and hence the lemma holds for such $n$ and $r$.

Computing the quantity on the right-hand side of (3.8) directly for $n=3$ and
$n=4$ and the remaining values of $r$ completes the proof of the lemma.

The lower bound of $1/(60n)$ is nearly attained for $n=3$, $m=1$ and $r=13$,
where the value of the right-hand side is $0.00565\ldots$. ∎

##  4\. The approximations and their bounds

We start by defining our sequence of approximations to $(a/b)^{m/n}$, along
with some estimates we will require.

Let $r$ be a non-negative integer, $a$ and $b$ be algebraic integers in an
imaginary quadratic field, $\mathbb{K}$, with either $0<b/a<1$ a rational
number or $|b/a|=1$ with $|b/a-1|<1$. Put $d=(a-b)^{2}$. Motivated by Lemma
3.2, we define

(4.1) $q_{r}=\frac{a^{r}D_{m,n,r}}{N_{d,m,n,r}}X_{m,n,r}(b/a)\hskip
8.53581pt\text{ and }\hskip
8.53581ptp_{r}=\frac{a^{r}D_{m,n,r}}{N_{d,m,n,r}}Y_{m,n,r}(b/a)=\frac{b^{r}D_{m,n,r}}{N_{d,m,n,r}}X_{m,n,r}(a/b).$

######  Lemma 4.1.

Let $r$ be a non-negative integer. Let $m$ and $n$ be relatively prime
positive integers with $0<m<n/2$. Then $p_{r}$ and $q_{r}$ are algebraic
integers with $p_{r}q_{r+1}\neq p_{r+1}q_{r}$ and

(4.2)
$\frac{D_{m,n,r}}{N_{d,m,n,r}}\left(|a|\left|1+\operatorname{Re}(b/a)\right|\right)^{r}\leq
q_{r}<1.072{\mathcal{C}}_{n}\left(\frac{{\mathcal{D}}_{n}}{{\mathcal{N}}_{d,n}}\right)^{r}\left|a^{1/2}+b^{1/2}\right|^{2r}.$

###### Proof.

The assertion that $p_{r}$ and $q_{r}$ are algebraic integers is just a
combination of our definitions of $p_{r}$, $q_{r}$, $D_{m,n,r}$ and
$N_{d,m,n,r}$.

That $p_{r}q_{r+1}\neq p_{r+1}q_{r}$ is equation (16) in Lemma 4 of [1].

We now prove the upper bound for $q_{r}$.

If $b/a$ is a rational number with $0<b<a$ relatively prime integers, then
from Lemma 5.2 of [14] (recalling again that $Y_{m,n,r}(z)$ there is
$X_{m,n,r}(z)$ here),

$a^{r}X_{m,n,r}(b/a)\leq{\left(a^{1/2}+b^{1/2}\right)}^{2r}.$

If $|b/a|=1$, then from Lemma 3.4,

$\left|\frac{a^{r}D_{m,n,r}}{N_{d,m,n,r}}X_{m,n,r}(b/a)\right|<1.072\frac{D_{m,n,r}}{N_{d,m,n,r}}\frac{r!\Gamma(1-m/n)}{\Gamma(r+1-m/n)}\left|\sqrt{a}+\sqrt{b}\right|^{2r}.$

The upper bound for $q_{r}$ now follows from this and (2.2).

The lower bound for $q_{r}$ is an immediate consequence of the lower bound for
$X_{m,n,r}(z)$ in Lemma 3.5. ∎

We next determine how close the resulting approximations are to $(a/b)^{m/n}$.

######  Lemma 4.2.

Let $m$ and $n$ be relatively prime positive integers with $m<n/2$ and $n\geq
3$. Let $r$ be a non-negative integer and let $a$ and $b$ be algebraic
integers in an imaginary quadratic field, $\mathbb{K}$, with either $0<b/a<1$
a rational number or $|b/a|=1$ with $|b/a-1|<1$. Then

(4.3)
$\left|\frac{a-b}{60naq_{r}}\right|<\left|q_{r}(a/b)^{m/n}-p_{r}\right|<1.22\left|\frac{a-b}{b}\right|{\mathcal{C}}_{n}\left(\frac{{\mathcal{D}}_{n}}{{\mathcal{N}}_{d,n}}\right)^{r}\left|\sqrt{a}-\sqrt{b}\right|^{2r}.$

###### Proof.

Using our definitions of $p_{r}$ and $q_{r}$ in (4.1), and of $R_{m,n,r}(z)$
in (3.4), along with the relation in (3.5) in Lemma 3.2 with $z=b/a$, we find
that

$\displaystyle p_{r}-q_{r}(a/b)^{m/n}$ $\displaystyle=$
$\displaystyle\left(\frac{a}{b}\right)^{m/n}\frac{a^{r}D_{m,n,r}}{N_{d,m,n,r}}{\left(\frac{b-a}{a}\right)}^{2r+1}\frac{(m/n)\cdots(r+m/n)}{(r+1)\cdots(2r+1)}$
$\displaystyle\times{}_{2}F_{1}\left(r+1-m/n,r+1;2r+2;(a-b)/a\right).$

Multiplying the top and bottom of the right-hand side by $q_{r}$, applying the
lower bound for $q_{r}$ in (4.2) in Lemma 4.1, along with (3.6) and then
simplifying, we obtain

$\left|q_{r}(a/b)^{m/n}-p_{r}\right|>\left|\left(\frac{D_{m,n,r}}{N_{d,m,n,r}}\right)^{2}\frac{(m/n)\cdots(r+m/n)}{(r+1)\cdots(2r+1)}(a-b)^{2r}(1+\operatorname{Re}(b/a))^{r}\frac{a-b}{aq_{r}}\right|.$

Recalling the definition of $N_{d,m,n,r}$, that $d=(a-b)^{2}$ and that
$X_{m,n,r}(z)$ is a monic polynomial, it follows that
$N_{d,m,n,r}\leq(a-b)^{r}$. Using this, along with Lemma 3.8, yields

$\left|q_{r}(a/b)^{m/n}-p_{r}\right|>\left|(1+\operatorname{Re}(b/a))^{r}\frac{a-b}{60naq_{r}}\right|.$

The desired lower bound now follows since $\operatorname{Re}(b/a)\geq 0$.

For the upper bound, we consider $b/a$ rational with $0<b/a<1$ and $|b/a|=1$
separately.

To obtain the upper bound when $|b/a|=1$, we start by writing
$b/a=\exp\left(\varphi i\right)$ with $-\pi<\varphi\leq\pi$ and showing that

(4.4) $|\varphi|\leq 1.22|(a-b)/b|$

for $|b/a-1|<1$.

If $|b/a-1|<1$, then $|\varphi|<1.05$. Using the well-known inequality
$2|\varphi|/\pi\leq|\sin(\varphi)|$ for all $|\varphi|\leq\pi$, we obtain
$|\varphi|<1.22|\sin(\varphi)|=1.22|\operatorname{Im}(b/a||$ for
$|\varphi|<1.05$. We can write $b/a=1-(a-b)/a$ and so
$\operatorname{Im}(b/a)=\operatorname{Im}(1-(a-b)/a)=-\operatorname{Im}((a-b)/a)$.
Since $|\operatorname{Im}(z)|\leq|z|$ for any complex number, $z$, equation
(4.4) follows.

We apply Lemma 2.5 of [4]:

$\left|R_{m,n,r}(b/a)\right|\leq\frac{\Gamma(r+1+m/n)}{r!\Gamma(m/n)}|\varphi|\left|1-\sqrt{b/a}\right|^{2r},$

where $|b/a-1|<1$ and $b/a=\exp\left(\varphi i\right)$. So

$\displaystyle\left|\left(\frac{a}{b}\right)^{m/n}\frac{a^{r}D_{m,n,r}}{N_{d,m,n,r}}R_{m,n,r}(b/a)\right|$
$\displaystyle\leq\frac{a^{r}D_{m,n,r}}{N_{d,m,n,r}}\frac{\Gamma(r+1+m/n)}{r!\Gamma(m/n)}|\varphi|\left|1-\sqrt{b/a}\right|^{2r}$
$\displaystyle<1.22\left|\frac{a-b}{b}\right|{\mathcal{C}}_{n}\left(\frac{{\mathcal{D}}_{n}}{{\mathcal{N}}_{d,n}}\right)^{r}\left|\sqrt{a}-\sqrt{b}\right|^{2r},$

by (4.4) and using (2.2).

To obtain the upper bound when $0<b/a<1$ is rational, we use Pochammer’s
integral (see equation (1.6.6) of [11]). We can write

${}_{2}F_{1}\left(r+1-m/n,r+1;2r+2;z\right)=\frac{\Gamma(2r+2)}{\Gamma(r+1)\Gamma(r+1)}\int_{0}^{1}t^{r}(1-t)^{r}(1-zt)^{-r-1+m/n}dt.$

If $0<z<1$, then $(1-zt)^{-1+m/n}$ is monotonically increasing as $t$ goes
from $0$ to $1$, since $m/n<1$, so its maximum value occurs at $t=1$. I.e.,
$(1-z)^{-1+m/n}\leq(1-z)^{-1+m/n}$. Here $z=1-b/a$, so this maximum is
$(b/a)^{-1+m/n}=(a/b)^{1-m/n}$.

Also, the function $t(1-t)(1-t(a-b)/a)^{-1}$ takes its maximum value at
$t=\sqrt{a}/\left(\sqrt{a}+\sqrt{b}\right)$, where it takes the value
$a/\left(\sqrt{a}+\sqrt{b}\right)^{2}$.

Hence

$\left|\int_{0}^{1}t^{r}(1-t)^{r}(1-zt)^{-r-1+m/n}dt\right|\leq(a/b)^{1-m/n}\left\\{a{\left(a^{1/2}+b^{1/2}\right)}^{-2}\right\\}^{r}$

and so

$\displaystyle\left|q_{r}(a/b)^{m/n}-p_{r}\right|$ $\displaystyle\leq$
$\displaystyle
a^{r}\frac{D_{m,n,r}}{N_{d,m,n,r}}{\left(\frac{a}{b}\right)}^{m/n}{\left(\frac{a-b}{a}\right)}^{2r+1}\frac{\Gamma(r+1+m/n)\Gamma(r+1)}{\Gamma(2r+2)\Gamma(m/n)}$
$\displaystyle\times\frac{\Gamma(2r+2)}{\Gamma(r+1)\Gamma(r+1)}(a/b)^{1-m/n}\left\\{a{\left(a^{1/2}+b^{1/2}\right)}^{-2}\right\\}^{r}$
$\displaystyle=$
$\displaystyle\frac{D_{m,n,r}}{N_{d,m,n,r}}{\left(\frac{a-b}{b}\right)}\frac{\Gamma(r+1+m/n)}{\Gamma(r+1)\Gamma(m/n)}\left(a^{1/2}-b^{1/2}\right)^{2r}$
$\displaystyle=$
$\displaystyle{\mathcal{C}}_{n}\left(\frac{{\mathcal{D}}_{n}}{{\mathcal{N}}_{d,n}}\right)^{r}\left(\frac{a-b}{b}\right)\left(a^{1/2}-b^{1/2}\right)^{2r},$

after simplifying and using (2.2). ∎

##  5\. Proof of Theorem 2.1

By the lower bound in Lemma 4.2, we can take $c$ in Theorem 2.1 to be $60n|a|$
when $p/q=p_{i}/q_{i}$ for some non-negative integer $i$. So we need only
prove Theorem 2.1 for those numbers $p/q\neq p_{i}/q_{i}$ for any non-negative
integer $i$.

All that is required is a simple application of Lemma 3.1 using Lemmas 4.1 and
4.2 to provide the values of $k_{0},\ell_{0},E$ and $Q$.

From these last two lemmas, we can choose $k_{0}=1.072{\mathcal{C}}_{n}$,
$E=\left({\mathcal{N}}_{d,n}/{\mathcal{D}}_{n}\right)\left|a^{1/2}-b^{1/2}\right|^{-2}$
and
$Q=\left({\mathcal{D}}_{n}/{\mathcal{N}}_{d,n}\right)\left|a^{1/2}+b^{1/2}\right|^{2}$.

From equation (4.3) in Lemma 4.2, an obvious choice for $\ell_{0}$ would be
$\ell_{0}=1.22|(a-b)/b|{\mathcal{C}}_{n}$. However, with this choice
$2\ell_{0}E<1$ if $|a-b|$ is not small, so the conditions in Lemma 3.1 are not
satisfied. We do not want to increase the size of $2\ell_{0}E$ too much, as
otherwise the dependence of $c$ and $a$ will increase. So we will choose
$\ell_{0}$ proportional to $1/E$, namely
$\ell_{0}=c_{1}{\mathcal{C}}_{n}{\mathcal{D}}_{n}\left|a^{1/2}-b^{1/2}\right|^{2}$
for some absolute constant $c_{1}>1$ such that
$\ell_{0}>1.22|(a-b)/b|{\mathcal{C}}_{n}$. With such a choice, the condition
$\left|q_{r}\theta-p_{r}\right|\leq\ell_{0}E^{-r}$ in Lemma 3.1 will hold and
$2\ell_{0}E=c_{1}n\mu_{n}{\mathcal{C}}_{n}\geq 1$ will also hold.

We first determine $c_{2}\geq 1$ such that

$\frac{|a-b|}{b}\leq
c_{2}\left|a^{1/2}-b^{1/2}\right|^{2}=c_{2}|b|\left|1-(a/b)^{1/2}\right|^{2}=c_{2}|b|\left|1-(1+(a-b)/b)^{1/2}\right|^{2}.$

If $a=a_{R}+a_{I}i$, then $(a-b)/b=2a_{I}i/\left(a_{R}-a_{I}i\right)$, so
$|(a-b)/b|\leq 2$. Using the series expansion of $\sqrt{1+z}$, we find that
$\left|1-\sqrt{1+z}\right|\geq\left(\sqrt{3}-1\right)|z|/2$ for all $|z|\leq
2$. Applying this inequality with $z=(a-b)/b$, we have

$\left|1-(1+(a-b)/b)^{1/2}\right|\geq\left(\sqrt{3}-1\right)|(a-b)/b|/2.$

So
$\left|a^{1/2}-b^{1/2}\right|^{2}\geq\left(\sqrt{3}-1\right)^{2}\left|(a-b)^{2}/(4b)\right|\geq\left(\sqrt{3}-1\right)^{2}\left|(a-b)/(4b)\right|$.

Hence we can take $c_{2}=2\left(2+\sqrt{3}\right)$ and put

$\ell_{0}=9.2{\mathcal{C}}_{n}{\mathcal{D}}_{n}\left|a^{1/2}-b^{1/2}\right|^{2},$

so that $\ell_{0}>1.22|(a-b)/b|{\mathcal{C}}_{n}$. Also,

$2\ell_{0}E=2\cdot
9.2{\mathcal{C}}_{n}{\mathcal{D}}_{n}\left|a^{1/2}-b^{1/2}\right|^{2}\left({\mathcal{N}}_{d,n}/{\mathcal{D}}_{n}\right)\left|a^{1/2}-b^{1/2}\right|^{-2}=18.4{\mathcal{C}}_{n}{\mathcal{N}}_{d,n}\geq
18.4.$

Lemma 4.1 assures us that $p_{r}q_{r+1}\neq p_{r+1}q_{r}$. In addition, as we
saw above, ${\mathcal{N}}_{d,n}\leq|a-b|$ and ${\mathcal{D}}_{n}\geq 1$, so
$Q\geq\left|a^{1/2}+b^{1/2}\right|^{2}/|a-b|>1$ and $\ell_{0}>0$ since $a\neq
b$. If $E>1$, then we can use Lemma 3.1.

Lastly, we consider the quantity $c$ in Lemma 3.1. Using the above expressions
for $2\ell_{0}E$, we can write it as

$2.15{\mathcal{C}}_{n}\left(18.4{\mathcal{C}}_{n}{\mathcal{N}}_{d,n}\right)^{\kappa}\leq
3|a|{\mathcal{C}}_{n}\left(20{\mathcal{C}}_{n}{\mathcal{N}}_{d,n}\right)^{\kappa},$

since ${\mathcal{N}}_{d,n}\leq n\mu_{n}$. Noting that $\kappa>1$, we see that
this is also larger than $60n|a|$, completing the proof of the Theorem 2.1.

##  6\. Proof of Theorem 2.2

For $n=3$, we use the bounds already established in Lemma 5.1(b) of [14].

The first two subsections provide the proof of parts (a) and (b) of the
theorem. We proceed in two steps for each $4\leq n\leq 757$.
1) we determine $r_{\text{comp}}$, such that for all $m$ and all $r\geq
r_{\text{comp}}$, we can use more-or-less analytic techniques to show that our
choice of ${\mathcal{D}}_{n}$ works with ${\mathcal{C}}_{n}=1$. So in this
step, we also determine the value of $cD_{n}$ we will use.
(2) for all $m$ and all $r<r_{\text{comp}}$, we essentially calculate directly
the quantity within the outer $\max$ on the left-hand side of equation (2.2).
It is in this way that we determine the value of ${\mathcal{C}}_{n}$ that we
need require.

We prove part (c) in the last subsection here.

###  6.1. Determining ${\mathcal{D}}_{n}$ and $r_{\text{comp}}$

We shall use estimates for each of the quantities on the left-hand side of
(2.2): the $\Gamma$ function quantities, the numerator and the denominator.

To estimate the denominator, $D_{m,n,r}$, we divide the prime divisors of
$D_{m,n,r}$ into two sets, according to their size. We let $D_{m,n,r}^{(S)}$
denote the contribution to $D_{m,n,r}$ from primes at most $(nr)^{1/2}$ and
let $D_{m,n,r}^{(L)}$ denote the contribution from the remaining, larger,
primes.

####  6.1.1. Numerator upper bounds

Put $d_{1}=\gcd\left(d,n^{2}\right)$ and
$d_{2}=\gcd\left(d/d_{1},n^{2}\right)$, as in [17]. By Lemma 6 of [17], we
have

$\frac{{\mathcal{N}}_{d,n}^{r}}{N_{d,m,n,r}}\leq\frac{\prod_{p\mid
n}p^{r\min(v_{p}(d)/2,v_{p}(n)+1/(p-1))}}{d_{1}^{\lfloor
r/2\rfloor}\prod_{p|d_{2}}p^{\min(\lfloor
v_{p}(d_{2})r/2\rfloor,v_{p}(r!))}}.$

We examine the terms in the products on the right-hand side and consider three
possibilities.

(i) If $v_{p}(n)>0$ and $v_{p}\left(d_{2}\right)=0$, then $v_{p}(d)/2\leq
v_{p}(n)$, so

$p^{r\min(v_{p}(d)/2,v_{p}(n)+1/(p-1))}=p^{rv_{p}(d)/2}=p^{rv_{p}(d_{1})/2}.$

(ii) If $p\geq 3$ and $p\mid d_{2}$, or if $p=2$ and
$v_{2}\left(d_{2}\right)\geq 2$, then
$p^{\min(v_{p}(d)/2,v_{p}(n)+1/(p-1))}=p^{v_{p}(n)+1/(p-1)}$. Furthermore, if
$p\geq 3$ and $p\mid d_{2}$, then $v_{p}(r!)\geq\min\left(\lfloor
v_{p}\left(d_{2}\right)r/2\rfloor,v_{p}(r!)\right)\geq\min\left(\lfloor
r/2\rfloor,v_{p}(r!)\right)=v_{p}(r!)$, since $v_{p}(r!)\leq r/(p-1)$.
Similarly, if $p=2$ and $v_{2}\left(d_{2}\right)\geq 2$, then
$v_{p}(r!)\geq\min\left(\lfloor
v_{p}\left(d_{2}\right)r/2\rfloor,v_{p}(r!)\right)\geq\min\left(\lfloor
r\rfloor,v_{p}(r!)\right)=v_{p}(r!)$. Thus

$\frac{p^{r\min(v_{p}(d)/2,v_{p}(n)+1/(p-1))}}{p^{\min(\lfloor
v_{p}(d_{2})r/2\rfloor,v_{p}(r!))}}=\frac{p^{r\left(v_{p}(n)+1/(p-1)\right)}}{p^{v_{p}(r!)}}=\frac{p^{r\left(v_{p}(d_{1})/2+1/(p-1)\right)}}{p^{v_{p}(r!)}},$

the last equality holding because
$v_{p}\left(d_{1}\right)v_{p}\left(n^{2}\right)$ when
$v_{p}\left(d_{2}\right)\geq 1$.

(iii) Lastly, if $p=2$ and $v_{p}\left(d_{2}\right)=1$, then

$\min\left(v_{p}(d)/2,v_{p}(n)+1/(p-1)\right)=\min\left(v_{p}\left(d_{1}\right)+1/2,v_{p}(n)+1\right).$

Since $v_{p}\left(d_{2}\right)>0$, it follows that
$\min\left(v_{p}\left(d_{1}\right)+1/2,v_{p}(n)+1\right)=v_{p}\left(d_{1}\right)+1/2$.
Also $\min\left(\lfloor
v_{p}\left(d_{2}\right)r/2\rfloor,v_{p}(r!)\right)=\min\left(\lfloor
r/2\rfloor,v_{p}(r!)\right)=\lfloor r/2\rfloor$. So

$\frac{p^{r\min(v_{p}(d)/2,v_{p}(n)+1/(p-1))}}{p^{\min(\lfloor
v_{p}(d_{2})r/2\rfloor,v_{p}(r!))}}=2^{rv_{2}(d_{1})/2}2^{r/2-\lfloor
r/2\rfloor}\leq 2^{rv_{2}(d_{1})/2}2^{r/(2-1)-v_{2}(r!)}.$

So we always have

(6.1)
$\frac{{\mathcal{N}}_{d,n}^{r}}{N_{d,m,n,r}}\leq\frac{d_{1}^{r/2}}{d_{1}^{\lfloor
r/2\rfloor}}\prod_{p|d_{2}}p^{r/(p-1)-v_{p}(r!)}.$

For $r\geq 1$, we have $0\leq r/(p-1)-v_{p}(r!)\leq(\log r)/(\log p)+1/(p-1)$
(the worst case being $r=1$). Therefore,

(6.2) $\frac{{\mathcal{N}}_{d,n}^{r}}{N_{d,m,n,r}}\leq n\mu_{n}r^{\omega(n)},$

where $\omega(n)$ is the number of distinct prime factors of $n$.

At least for $r=1$, there are examples showing that this upper bound is sharp.
For larger $r$, it can also be not bad.

####  6.1.2. $\Gamma$-term upper bounds

When considering the $\Gamma$-term estimates in the proof of Lemma 7.4(c0 of
[15], we showed that

$\max\left(1,\frac{\Gamma(1-m/n)\,r!}{\Gamma(r+1-m/n)},\frac{n\Gamma(r+1+m/n)}{m\Gamma(m/n)r!}\right)\frac{{\mathcal{N}}_{d,n}^{r}}{N_{d,m,n,r}}<\frac{n}{n-m}e^{m^{2}/n^{2}}r^{m/n},$

for $n\geq 2$. Since $m<n/2$, we have

(6.3)
$\max\left(1,\frac{\Gamma(1-m/n)\,r!}{\Gamma(r+1-m/n)},\frac{n\Gamma(r+1+m/n)}{m\Gamma(m/n)r!}\right)\frac{{\mathcal{N}}_{d,n}^{r}}{N_{d,m,n,r}}<(n/2)e^{1/4}r^{1/2}.$

####  6.1.3. $D_{m,n,r}^{(S)}$ upper bounds

From Lemma 3.3(a) of [14], we know that

$D_{m,n,r}^{(S)}\leq\prod_{(nr)^{1/3}<p\leq(nr)^{1/2}}p^{2}\prod_{(nr)^{1/4}<p\leq(nr)^{1/3}}p^{3}\prod_{p\leq(nr)^{1/4}}p^{\lfloor\log(nr)/(\log(p))\rfloor}.$

So

$\log D_{m,n,r}^{(S)}\leq
2\theta\left((nr)^{1/2}\right)+\theta\left((nr)^{1/3}\right)-3\theta\left((nr)^{1/4}\right)+\sum_{p\leq(nr)^{1/4}}\lfloor\log(nr)/(\log(p))\rfloor\log(p).$

Now $\lfloor x\rfloor\leq 4\lfloor x/4\rfloor+3$, so

$\sum_{p\leq(nr)^{1/4}}\lfloor\log(nr)/(\log(p))\rfloor\log(p)\leq
4\psi\left((nr)^{1/4}\right)+3\theta\left((nr)^{1/4}\right),$

where $\displaystyle\theta(x)=\sum_{\stackrel{{\scriptstyle p\leq
x}}{{p,\text{ prime}}}}\log(p)$ and
$\displaystyle\psi(x)=\sum_{\stackrel{{\scriptstyle p^{n}\leq x}}{{p,\text{
prime}}}}\log(p)$.

Thus,

(6.4) $\displaystyle D_{m,n,r}^{(S)}$ $\displaystyle\leq$
$\displaystyle\exp\left\\{2\theta\left((nr)^{1/2}\right)+\theta\left((nr)^{1/3}\right)+4\psi\left((nr)^{1/4}\right)\right\\}$
$\displaystyle<$
$\displaystyle\exp\left\\{2.033(nr)^{1/2}+1.017(nr)^{1/3}+4.156(nr)^{1/4}\right\\},$

from Theorems 9 and 12 of [9].

From equations (6.2), (6.3) and (6.4), we obtain

(6.5)
$\displaystyle\max\left(1,\frac{\Gamma(1-m/n)\,r!}{\Gamma(r+1-m/n)},\frac{n\Gamma(r+1+m/n)}{m\Gamma(m/n)r!}\right)\frac{{\mathcal{N}}_{d,n}^{r}}{N_{d,m,n,r}}D_{m,n,r}^{(S)}$
$\displaystyle<$ $\displaystyle
0.65n^{2}\mu_{n}r^{\omega(n)+1/2}\exp\left\\{2.033(nr)^{1/2}+1.017(nr)^{1/3}+4.156(nr)^{1/4}\right\\}.$

For convenience in what follows, we will denote this last quantity as
$S(n,r)$.

####  6.1.4. Upper and lower bounds for $\theta(x;n,k)$

To obtain an upper bound for $D_{m,n,r}^{(L)}$, we need upper and lower bounds
for $\theta\left(x;n,k\right)$. We want bounds of the form
$\epsilon_{x,n,k}^{(L)}x<\theta(x;n,k)-x/\varphi(n)<\epsilon_{x,n,k}^{(U)}x$.
For this, we use the results in [3] and some computation.

Combining equations (3.7) and (3.8) in Lemma 3.6, we find that the upper bound
for $\left|\theta(x;n,k)-x/\varphi(n)\right|$ in (3.7) holds for $x\geq
1.8\cdot 10^{9}$ when $3\leq n\leq 10^{4}$ and for $x\geq 2.1\cdot 10^{9}$
when $4\leq n\leq 100$.

For each $4\leq n\leq 757$, we compute $\theta(x;n,k)$ for all $1\leq k<n$
with $\gcd(k,n)=1$ and for all $x\leq 2.1\cdot 10^{9}$ to find the last value
of $x$, $X_{n}$, that breaches (3.7) for any $1\leq k<n$ with $\gcd(k,n)=1$.

However, these $X_{n}$’s are still quite large (e.g., $X_{4}=1,472,117,809$),
which means that $r$ would have to be quite large for (3.7) to give a good
upper bound for $D_{m,n,r}^{(L)}$. So we break the interval
$\left[1,X_{n}+2000\right]$ into $\lfloor X_{n}/2000\rfloor+2$ subintervals of
size $2000$, $I_{i}=[2000(i-1)+1,2000i]$, and compute to obtain values
$\epsilon_{LB,i}$ and $\varepsilon_{UB,i}$ such that if $x\geq 2000(i-1)+1$,
then $\epsilon_{LB,i}x<\theta(x;n,k)-x/\varphi(n)<\epsilon_{UB,i}x$ for all
$1\leq k\leq n$ with $\gcd(k,n)=1$.

For any positive real number $x$, let $i$ be the largest positive integer such
that $x\geq 2000(i-1)+1$. we will let
$\theta_{UB}(x;n)=x/\varphi(n)+\epsilon_{UB,i}x$ and
$\theta_{LB}(x;n)=x/\varphi(n)-\epsilon_{LB,i}x$. Note that
$\theta_{UB}(x;n)>\theta(x;n,k)>\theta_{LB}(x;n)$ for all $1\leq k\leq n$ with
$\gcd(k,n)=1$. This notation will be convenient for us in what follows.

####  6.1.5. $D_{m,n,r}^{(L)}$ upper bounds

From Lemma 3.3(b) of [14], we see that for any positive integer $N$ satisfying
$nr/(nN+n/2)\geq(nr)^{1/2}$, we have

$\displaystyle D_{m,n,r}^{(L)}$ $\displaystyle\leq$
$\displaystyle\exp\left\\{\sum_{A=0}^{N-1}\sum_{\ell=1,(\ell,n)=1}^{n/2}\left(\theta(nr/(nA+\ell);n,k_{\ell})-\theta(nr/(nA+n-\ell);n,k_{\ell})\right)\right\\}$
$\displaystyle\times\exp\left\\{\sum_{\ell=1,(\ell,n)=1}^{n/2}\theta(nr/(nN+\ell);n,k_{\ell})\right\\}$
$\displaystyle<$
$\displaystyle\exp\left\\{\sum_{A=0}^{N-1}\sum_{\ell=1,(\ell,n)=1}^{n/2}\left(\theta_{UB}(nr/(nA+\ell);n)-\theta_{LB}(nr/(nA+n-\ell);n)\right)\right\\}$
$\displaystyle\times\exp\left\\{\sum_{\ell=1,(\ell,n)=1}^{n/2}\theta_{UB}(nr/(nN+\ell);n)\right\\},$

where $k_{\ell}\equiv(-m)\ell^{-1}\bmod n$. We will denote the last quantity
as $D^{(L)}(N,n,r)$.

####  6.1.6. Combining the bounds

Combining equations (6.5) and (6.1.5), we find that the left-hand side of
(2.2) is less than $S(n,r)D^{(L)}(N,n,r)$.

Incrementing $r$ in steps of size $100,000$ and checking positive integers $N$
up to $200$, we determined $\log\left(S(n,r)D^{(L)}(N,n,r)\right)$ for each
pair $(r,N)$ and then chose the values of $r$ and $N$ (we denote this $r$ by
$r_{{\rm comp}}$) such that $\log\left(S(n,r)D^{(L)}(N,n,r)\right)/r$ is as
small as possible to obtain an upper bound for left-hand side of (2.2) once
$r\geq r_{{\rm comp}}$. This is the value we will use for
$\log{\mathcal{D}}_{n}$. E.g., for $n=4$, $N=90$ and $r_{\rm comp}=39,900,000$
suggests using $\log{\mathcal{D}}_{3}=1.58$.

We now know ${\mathcal{D}}_{n}$ as well as how much computation is required to
establish our desired inequalities for all $r\geq 0$ (a computation which will
yield ${\mathcal{C}}_{n}$), so we are ready to describe the required
computations.

###  6.2. Determining ${\mathcal{C}}_{n}$ and checking $r<r_{\text{comp}}$

For each pair $(m,n)$ with $1\leq m<n/2$, $4\leq n\leq 757$ and $\gcd(m,n)=1$,
we take the following steps for each $0\leq r<r_{{\rm comp}}$.

(1) We compute directly the $\Gamma$ terms in (2.2), noting that the value for
$r$ can be computed from the value for $r+1$.

(2) We initially estimate the numerator, ${\mathcal{N}}_{d,n}^{r}/N_{d,m,n,r}$
in fact, using (6.1), where we bound $d_{1}^{r/2-\lfloor r/2\rfloor}$ from
above by $n$ and take the product over all primes dividing $n$, rather than
$d_{2}$.

This is much faster than calculating the maximum possible value of
${\mathcal{N}}_{d,n}^{r}/N_{d,m,n,r}$ precisely over all values of $d$.
However, if, for a particular value of $r$, after the denominator steps that
follow, this estimate leads to a large value of ${\mathcal{C}}_{n}$, then we
do calculate ${\mathcal{N}}_{d,n}^{r}/N_{d,m,n,r}$ more precisely using the
expression for $X_{m,n,r}\left(1-\sqrt{d}\,x\right)$ in terms of $d_{1}$,
$d_{2}$ and $d_{3}$ in the proof of Lemma 6 in [17].

(3) we initially use the upper bound

$D_{m,n,r}^{(S)}\leq\prod_{p\leq(nr)^{1/2}}p^{\lfloor\log(nr)/(\log(p))\rfloor},$

which holds by Lemma 3.3(a) of [14]. We calculate the right-hand side directly
for each value of $m$, $n$ and $r$.

As in step (2), if this upper bound leads to a large value of
${\mathcal{C}}_{n}$, then we calculate $D^{(S)}_{m,n,r}$ directly using
Proposition 3.2 of [14].

(4) we compute $D^{(L)}_{m,n,r}$ exactly using the same technique as in [14]
(see Step (5) of the proof of Lemma 5.1(b) there) of using Lemma 3.3(b) there
and calculating the contributions from each interval and congruence class via
the endpoints of these intervals. The only difference is that here we grow
what is called $A(r)$ in [14] over the course of the calculation so that
$A(r)$ is the largest integer such that $nr/(nA(r)+n-\ell>\sqrt{nr}$.

In this manner, for all $r<r_{{\rm comp}}$, we estimate the left-hand side of
(2.2) and hence find a value of ${\mathcal{C}}_{n}$ that would work with the
value of ${\mathcal{D}}_{n}$. If for any such $r$, the value of
${\mathcal{C}}_{n}$ exceeds the value of ${\mathcal{C}}_{n}$ found for smaller
values of $r$, then we use the more precise methods for bounding
${\mathcal{N}}_{d,n}^{r}/N_{d,m,n,r}$ and $D^{(S)}(m,n,r)$ described in steps
(2) and (3) above to get a more precise upper bound for ${\mathcal{C}}_{n}$.
So the maximum value of ${\mathcal{C}}_{n}$ obtained in this way is the one
that we use.

As part of these calculations, we also determined ${\mathcal{D}}_{2,n}$ in
Tables 1 and 2.

All these calculations were performed using code written in the Java
programming language (JDK 16). The code is available upon request.

###  6.3. Proof of Theorem 2.2(c)

It was shown in the proof of Lemma 7.4(d) of [15] that

$\max\left(1,\frac{\Gamma(1-m/n)\,r!}{\Gamma(r+1-m/n)},\frac{n\Gamma(r+1+m/n)}{m\Gamma(m/n)r!}\right)D_{m,n,r}\leq
n^{r}\mu_{n}^{r}.$

Applying equation (6.1), if $d_{2}=1$, then part (c) follows as $d_{1}\leq
n^{2}$.

##  7\. Thue’s Fundamentaltheorem

The initial hope for this work was to improve the constant not just for the
usual hypergeometric method, but for Thue’s Fundamentaltheorem too (e.g.,
Theorem 1 in [17], as well as the theorems in [15, 16]).

The two key parts of Thue’s Fundamentaltheorem are the following.

(1) let $t$ be a rational integer which is not a perfect square and put
$\mathbb{K}=\mathbb{Q}\left(\sqrt{t}\right)$. Suppose that
$\eta\in{\mathcal{O}}_{\mathbb{K}}$ and that $\sigma$ is the non-trivial
element of $\operatorname{Gal}\left(\mathbb{K}/\mathbb{Q}\right)$. Then
$\sigma(\eta)^{r}X_{m,n,r}\left(\eta/\sigma(\eta)\right)$ and
$\sigma(\eta)^{r}Y_{m,n,r}\left(\eta/\sigma(\eta)\right)$ are algebraic
conjugates in $\mathbb{K}$.

(2) the classical observation (see Lemma 3.2) that

$\left(\eta/\sigma(\eta)\right)^{m/n}Y_{m,n,r}\left(\eta/\sigma(\eta)\right)-X_{m,n,r}\left(\eta/\sigma(\eta)\right)=\left(\eta/\sigma(\eta)-1\right)^{2r+1}R_{m,n,r}\left(\eta/\sigma(\eta)\right),$

for $\left|\eta/\sigma(\eta)-1\right|<1$.

Due to (1), we can write

$\left(\eta/\sigma(\eta)\right)^{m/n}\sigma\left(\sigma(\eta)^{r}X_{m,n,r}\left(\eta/\sigma(\eta)\right)\right)-\sigma(\eta)^{r}X_{m,n,r}\left(\eta/\sigma(\eta)\right)=\left(\eta/\sigma(\eta)-1\right)^{2r+1}\sigma(\eta)^{r}R_{m,n,r}\left(\eta/\sigma(\eta)\right),$

To simplify our notation, we will write
$q_{r}=\sigma\left(\sigma(\eta)^{r}X_{m,n,r}\left(\eta/\sigma(\eta)\right)\right)$.

Let $\beta$ and $\gamma$ be two distinct non-rational algebraic integers in
$\mathbb{K}$ and put

$\alpha=\frac{\beta+\sigma(\beta)\left(\eta/\sigma(\eta)\right)^{m/n}}{\gamma+\sigma(\gamma)\left(\eta/\sigma(\eta)\right)^{m/n}}.$

Using the idea from [16], we can write

$\left(\gamma
q_{r}+\sigma(\gamma)\sigma\left(q_{r}\right)\right)\alpha-\left(q_{r}\beta+\sigma(\beta)\sigma\left(q_{r}\right)\right)=\left(\sigma(\beta)-\alpha\sigma(\gamma)\right)\left(\eta/\sigma(\eta)-1\right)^{2r+1}\sigma(\eta)^{r}R_{m,n,r}\left(\eta/\sigma(\eta)\right).$

This gives us a sequence of good approximations to $\alpha$ from our sequence
of good approximations to $\left(\eta/\sigma(\eta)\right)^{m/n}$.

As above with the usual hypergeometric method, to get improved constants we
need a lower bound for $\left|\gamma
q_{r}+\sigma(\gamma)\sigma\left(q_{r}\right)\right|$. Notice that if $\gamma
q_{r}=a_{r}+b_{r}\sqrt{t}$, then $\gamma
q_{r}+\sigma(\gamma)\sigma\left(q_{r}\right)=2a_{r}$. How can we bound
$\left|2a_{r}\right|$ from below?

Unfortunately, it is easy to compute examples with the real parts of the
values of the above hypergeometric functions having sign changes on the unit
circle that get closer to $1$ as $r$ gets larger. This seems to suggest that
our approach here will not provide better constants for Thue’s
Fundamentaltheorem. But perhaps it is only some fresh ideas that are required.

## Acknowledgements

Some of the ideas in this paper resulted from discussions during the
“Transcendence and Diophantine Problems” conference celebrating 100 years
since Feldman’s birth, held at MIPT, Moscow, 10–14 June 2019. The author
thanks Professor Moshchevitin for the invitation, as well as for his
generosity and assistance during this conference, and for providing a
stimulating environment.

##  Appendix A Values of ${\mathcal{C}}_{n}$, ${\mathcal{D}}_{n}$ and
supporting data

In the following tables, we provide the values of ${\mathcal{C}}_{1,n}$,
$\log{\mathcal{D}}_{1,n}$ and $\log{\mathcal{D}}_{2,n}$ for parts (a) and (b)
of Theorem 2.2. We also provide information about the calculations used in the
proof of this theorem, as described in Section 6.

Here is a description of the other fields in these tables.
$\bullet$ $m_{1,\text{max}}$: the value of $m$ where the maximum value of
${\mathcal{C}}_{1,n}$ occurred.
$\bullet$ $\log{\mathcal{D}}_{\text{Chud},n}$: the value of
${\mathcal{D}}_{n}$ for Chudnovsky’s asymptotic estimate.
$\bullet$ $\log n\mu_{n}$: the value of ${\mathcal{D}}_{n}$ used by Baker and
defined in Theorem 2.2.
These two values are provided for comparison with our own values.
$\bullet$ $r_{1,\text{max}}$: the value of $r$ where the maximum value of
${\mathcal{C}}_{1,n}$ occurred.
$\bullet$ $m_{2,\text{max}}$: the value of $m$ where ${\mathcal{C}}_{n}=100$
with ${\mathcal{D}}_{n}={\mathcal{D}}_{2,n}$ occurred.
$\bullet$ $r_{2,\text{max}}$: the value of $r$ where ${\mathcal{C}}_{n}=100$
with ${\mathcal{D}}_{n}={\mathcal{D}}_{2,n}$ occurred.
$\bullet$ $r_{\text{comp}}$: defined at the start of Section 6.

Note that $m_{1,\text{max}}$ and $m_{2,\text{max}}$ are not included in Tables
4–6, since we only consider $m=1$ for such values of $n$.

In some cases, especially for large $n$, ${\mathcal{C}}_{n}=100$ is never
attained with the values of ${\mathcal{D}}_{n}$ that we can use. In these
cases, the entries of $m_{2,\text{max}}$ and $r_{2,\text{max}}$ in the tables
are “–”.

$n$ | ${\mathcal{C}}_{1,n}$ | $\log{\mathcal{D}}_{\text{Chud},n}$ | $\log{\mathcal{D}}_{1,n}$ | $\log{\mathcal{D}}_{2,n}$ | $\log n\mu_{n}$ | $m_{1,\text{max}}$ | $m_{2,\text{max}}$ | $r_{1,\text{max}}$ | $r_{2,\text{max}}$ | $r_{\text{comp}}$ | $N$
---|---|---|---|---|---|---|---|---|---|---|---
3 | $2\cdot 10^{14}$ | 0.907 | 0.916 | 0.953 | 1.648 | 1 | 1 | 19,946 | 66 | $200,000,000$ | 201
4 | $3\cdot 10^{26}$ | 1.571 | 1.579 | 1.635 | 2.080 | 1 | 1 | 14,983 | 165 | $49,600,000$ | 99
5 | $10^{45}$ | 1.337 | 1.348 | 1.410 | 2.012 | 1 | 2 | 7060 | 200 | $44,800,000$ | 77
6 | $7\cdot 10^{24}$ | 2.721 | 2.729 | 2.761 | 3.035 | 1 | 1 | 9912 | 271 | $36,000,000$ | 65
7 | $10^{26}$ | 1.625 | 1.638 | 1.716 | 2.271 | 1 | 2 | 12364 | 293 | $46,300,000$ | 65
8 | $8\cdot 10^{20}$ | 2.222 | 2.235 | 2.348 | 2.773 | 1 | 1 | 3529 | 61 | $40,900,000$ | 57
9 | $5\cdot 10^{31}$ | 2.155 | 2.169 | 2.288 | 2.747 | 1 | 2 | 13,953 | 52 | $39,700,000$ | 58
10 | $2\cdot 10^{26}$ | 2.988 | 2.999 | 3.064 | 3.399 | 1 | 1 | 2383 | 107 | $40,400,000$ | 52
11 | $7\cdot 10^{23}$ | 2.020 | 2.038 | 2.158 | 2.638 | 5 | 5 | 2161 | 114 | $43,400,000$ | 45
12 | $3\cdot 10^{32}$ | 3.142 | 3.155 | 3.258 | 3.728 | 5 | 1 | 3568 | 42 | $48,000,000$ | 56
13 | $4\cdot 10^{24}$ | 2.169 | 2.189 | 2.314 | 2.779 | 4 | 4 | 3234 | 30 | $45,700,000$ | 38
14 | $2\cdot 10^{30}$ | 3.203 | 3.216 | 3.350 | 3.657 | 3 | 3 | 2794 | 47 | $45,400,000$ | 55
15 | $7\cdot 10^{30}$ | 3.125 | 3.141 | 3.283 | 3.660 | 4 | 7 | 12515 | 61 | $45,700,000$ | 45
16 | $3\cdot 10^{51}$ | 2.903 | 2.920 | 3.061 | 3.466 | 3 | 3 | 7759 | 55 | $49,000,000$ | 48
17 | $4\cdot 10^{22}$ | 2.410 | 2.435 | 2.576 | 3.011 | 4 | 8 | 2424 | 23 | $49,400,000$ | 35
18 | $3\cdot 10^{26}$ | 3.600 | 3.613 | 3.713 | 4.133 | 1 | 5 | 1553 | 113 | $48,700,000$ | 59
19 | $4\cdot 10^{20}$ | 2.511 | 2.538 | 2.741 | 3.109 | 1 | 6 | 2806 | 73 | $47,200,000$ | 28
20 | $5\cdot 10^{23}$ | 3.513 | 3.530 | 3.666 | 4.092 | 3 | 1 | 4061 | 36 | $44,500,000$ | 43
21 | $4\cdot 10^{32}$ | 3.375 | 3.395 | 3.527 | 3.919 | 4 | 8 | 1507 | 183 | $44,200,000$ | 35
22 | $5\cdot 10^{27}$ | 3.530 | 3.548 | 3.666 | 4.024 | 7 | 7 | 3283 | 107 | $42,500,000$ | 42
23 | $7\cdot 10^{17}$ | 2.687 | 2.715 | 2.908 | 3.279 | 8 | 10 | 1579 | 73 | $48,400,000$ | 27
24 | $7\cdot 10^{37}$ | 3.848 | 3.868 | 3.997 | 4.421 | 11 | 11 | 5920 | 102 | $47,400,000$ | 37
25 | $2\cdot 10^{28}$ | 3.049 | 3.077 | 3.280 | 3.622 | 8 | 8 | 3252 | 52 | $45,000,000$ | 28
26 | $8\cdot 10^{26}$ | 3.660 | 3.680 | 3.792 | 4.165 | 7 | 9 | 1984 | 165 | $48,100,000$ | 34
27 | $4\cdot 10^{19}$ | 3.275 | 3.303 | 3.453 | 3.846 | 8 | 4 | 1251 | 27 | $44,800,000$ | 28
28 | $5\cdot 10^{25}$ | 3.774 | 3.796 | 3.993 | 4.350 | 3 | 13 | 2018 | 38 | $46,400,000$ | 34
29 | $8\cdot 10^{20}$ | 2.901 | 2.936 | 3.185 | 3.488 | 3 | 3 | 601 | 29 | $47,000,000$ | 22
30 | $5\cdot 10^{39}$ | 4.431 | 4.449 | 4.592 | 5.047 | 7 | 7 | 2093 | 102 | $46,000,000$ | 48
31 | $4\cdot 10^{24}$ | 2.963 | 3.000 | 3.216 | 3.549 | 14 | 12 | 1496 | 31 | $49,200,000$ | 22
32 | $2\cdot 10^{27}$ | 3.593 | 3.619 | 3.821 | 4.159 | 7 | 15 | 1231 | 44 | $49,500,000$ | 29
33 | $2\cdot 10^{29}$ | 3.734 | 3.761 | 3.900 | 4.286 | 4 | 8 | 1550 | 23 | $48,800,000$ | 29
34 | $4\cdot 10^{35}$ | 3.877 | 3.903 | 4.013 | 4.397 | 11 | 3 | 2642 | 59 | $46,100,000$ | 31
35 | $3\cdot 10^{27}$ | 3.730 | 3.760 | 3.960 | 4.283 | 1 | 9 | 2470 | 58 | $47,200,000$ | 26
36 | $6\cdot 10^{56}$ | 4.256 | 4.278 | 4.427 | 4.826 | 7 | 17 | 5305 | 16 | $49,100,000$ | 38
37 | $5\cdot 10^{20}$ | 3.129 | 3.169 | 3.352 | 3.712 | 11 | 18 | 1009 | 17 | $49,600,000$ | 19
38 | $4\cdot 10^{16}$ | 3.970 | 3.997 | 4.152 | 4.495 | 15 | 3 | 909 | 67 | $47,500,000$ | 28
39 | $5\cdot 10^{38}$ | 3.873 | 3.904 | 4.064 | 4.427 | 19 | 19 | 6609 | 94 | $46,800,000$ | 28
40 | $9\cdot 10^{39}$ | 4.214 | 4.242 | 4.364 | 4.785 | 19 | 3 | 1809 | 32 | $48,500,000$ | 28
41 | $7\cdot 10^{21}$ | 3.226 | 3.270 | 3.448 | 3.807 | 15 | 5 | 907 | 43 | $47,000,000$ | 19
42 | $9\cdot 10^{35}$ | 4.703 | 4.724 | 4.912 | 5.305 | 19 | 13 | 5452 | 25 | $49,400,000$ | 40
43 | $2\cdot 10^{19}$ | 3.271 | 3.316 | 3.535 | 3.851 | 4 | 10 | 1596 | 45 | $45,900,000$ | 18
44 | $3\cdot 10^{28}$ | 4.145 | 4.175 | 4.316 | 4.718 | 7 | 15 | 4890 | 55 | $45,900,000$ | 26

Table 1. Data for $3\leq n\leq 44$

$n$ | ${\mathcal{C}}_{1,n}$ | $\log{\mathcal{D}}_{\text{Chud},n}$ | $\log{\mathcal{D}}_{1,n}$ | $\log{\mathcal{D}}_{2,n}$ | $\log n\mu_{n}$ | $m_{1,\text{max}}$ | $m_{2,\text{max}}$ | $r_{1,\text{max}}$ | $r_{2,\text{max}}$ | $r_{\text{comp}}$ | $N$
---|---|---|---|---|---|---|---|---|---|---|---
45 | $5\cdot 10^{20}$ | 4.196 | 4.228 | 4.388 | 4.759 | 22 | 11 | 1480 | 66 | $48,100,000$ | 24
46 | $4\cdot 10^{20}$ | 4.133 | 4.162 | 4.314 | 4.665 | 5 | 19 | 1615 | 68 | $48,600,000$ | 26
47 | $4\cdot 10^{21}$ | 3.355 | 3.402 | 3.589 | 3.934 | 5 | 19 | 1631 | 18 | $47,000,000$ | 17
48 | $10^{22}$ | 4.545 | 4.573 | 4.751 | 5.114 | 7 | 7 | 3982 | 32 | $46,900,000$ | 30
49 | $2\cdot 10^{19}$ | 3.647 | 3.691 | 3.849 | 4.217 | 24 | 8 | 688 | 39 | $48,400,000$ | 20
50 | $9\cdot 10^{26}$ | 4.448 | 4.477 | 4.604 | 5.008 | 7 | 7 | 3503 | 102 | $50,000,000$ | 29
51 | $2\cdot 10^{20}$ | 4.101 | 4.139 | 4.345 | 4.659 | 25 | 22 | 1135 | 27 | $49,100,000$ | 22
52 | $2\cdot 10^{27}$ | 4.287 | 4.320 | 4.460 | 4.859 | 3 | 9 | 1712 | 192 | $44,900,000$ | 26
53 | $4\cdot 10^{17}$ | 3.469 | 3.524 | 3.708 | 4.047 | 17 | 6 | 762 | 45 | $41,900,000$ | 14
54 | $5\cdot 10^{20}$ | 4.668 | 4.697 | 4.885 | 5.232 | 11 | 7 | 2062 | 89 | $47,700,000$ | 28
55 | $8\cdot 10^{31}$ | 4.092 | 4.135 | 4.296 | 4.650 | 4 | 14 | 567 | 27 | $46,400,000$ | 18
56 | $2\cdot 10^{27}$ | 4.473 | 4.508 | 4.706 | 5.043 | 1 | 19 | 587 | 105 | $47,300,000$ | 23
57 | $6\cdot 10^{26}$ | 4.198 | 4.240 | 4.459 | 4.756 | 23 | 28 | 1437 | 27 | $47,900,000$ | 17
58 | $5\cdot 10^{27}$ | 4.335 | 4.371 | 4.568 | 4.874 | 17 | 17 | 722 | 33 | $47,900,000$ | 21
59 | $2\cdot 10^{18}$ | 3.571 | 3.630 | 3.957 | 4.148 | 24 | 28 | 655 | 27 | $37,700,000$ | 13
60 | $3\cdot 10^{26}$ | 5.176 | 5.203 | 5.388 | 5.740 | 19 | 11 | 2171 | 96 | $48,000,000$ | 27
61 | $5\cdot 10^{19}$ | 3.603 | 3.664 | 3.876 | 4.180 | 17 | 8 | 1096 | 21 | $35,900,000$ | 13
62 | $10^{22}$ | 4.394 | 4.433 | 4.660 | 4.935 | 23 | 23 | 2398 | 31 | $46,500,000$ | 20
63 | $3\cdot 10^{21}$ | 4.453 | 4.494 | 4.723 | 5.017 | 10 | 29 | 589 | 27 | $50,000,000$ | 21
64 | $3\cdot 10^{31}$ | 4.285 | 4.326 | 4.476 | 4.853 | 31 | 9 | 1711 | 47 | $47,100,000$ | 20
65 | $3\cdot 10^{22}$ | 4.232 | 4.281 | 4.505 | 4.791 | 14 | 28 | 677 | 27 | $43,100,000$ | 17
66 | $9\cdot 10^{25}$ | 5.082 | 5.112 | 5.267 | 5.672 | 19 | 29 | 1383 | 35 | $48,000,000$ | 30
67 | $4\cdot 10^{16}$ | 3.693 | 3.759 | 3.923 | 4.269 | 17 | 27 | 635 | 134 | $31,500,000$ | 13
68 | $2\cdot 10^{29}$ | 4.519 | 4.560 | 4.752 | 5.090 | 21 | 31 | 1564 | 67 | $49,600,000$ | 20
69 | $4\cdot 10^{15}$ | 4.366 | 4.412 | 4.623 | 4.926 | 1 | 7 | 707 | 26 | $48,100,000$ | 19
70 | $3\cdot 10^{26}$ | 5.080 | 5.112 | 5.287 | 5.669 | 23 | 17 | 1120 | 31 | $49,400,000$ | 27
71 | $2\cdot 10^{17}$ | 3.749 | 3.819 | 4.073 | 4.324 | 10 | 14 | 1096 | 13 | $29,700,000$ | 11
72 | $4\cdot 10^{27}$ | 4.951 | 4.987 | 5.160 | 5.520 | 31 | 35 | 4221 | 124 | $48,000,000$ | 21
73 | $5\cdot 10^{13}$ | 3.775 | 3.848 | 4.053 | 4.351 | 11 | 4 | 442 | 31 | $29,200,000$ | 11
74 | $3\cdot 10^{20}$ | 4.553 | 4.595 | 4.807 | 5.098 | 27 | 1 | 1549 | 45 | $48,000,000$ | 18
75 | $6\cdot 10^{24}$ | 4.704 | 4.748 | 4.967 | 5.270 | 2 | 8 | 1913 | 58 | $49,200,000$ | 21
76 | $9\cdot 10^{32}$ | 4.617 | 4.659 | 4.874 | 5.188 | 23 | 31 | 446 | 57 | $48,500,000$ | 18
77 | $2\cdot 10^{19}$ | 4.348 | 4.409 | 4.600 | 4.908 | 3 | 15 | 576 | 101 | $35,100,000$ | 13
78 | $9\cdot 10^{23}$ | 5.227 | 5.261 | 5.493 | 5.813 | 19 | 31 | 1841 | 92 | $44,900,000$ | 28
79 | $3\cdot 10^{11}$ | 3.851 | 3.928 | 4.150 | 4.426 | 25 | 12 | 101 | 11 | $27,800,000$ | 10
80 | $9\cdot 10^{28}$ | 4.910 | 4.950 | 5.126 | 5.478 | 3 | 33 | 1571 | 107 | $47,700,000$ | 20
81 | $3\cdot 10^{18}$ | 4.376 | 4.435 | 4.658 | 4.944 | 40 | 23 | 484 | 50 | $39,400,000$ | 14
82 | $9\cdot 10^{18}$ | 4.646 | 4.692 | 4.900 | 5.193 | 35 | 35 | 822 | 37 | $48,400,000$ | 19
83 | $3\cdot 10^{21}$ | 3.899 | 3.980 | 4.172 | 4.473 | 18 | 8 | 765 | 23 | $26,400,000$ | 10
84 | $3\cdot 10^{21}$ | 5.433 | 5.468 | 5.643 | 5.998 | 37 | 23 | 1017 | 94 | $49,000,000$ | 24
85 | $4\cdot 10^{15}$ | 4.461 | 4.527 | 4.713 | 5.023 | 19 | 9 | 568 | 17 | $32,600,000$ | 12
86 | $3\cdot 10^{17}$ | 4.689 | 4.736 | 4.919 | 5.238 | 17 | 11 | 593 | 53 | $49,200,000$ | 18

Table 2. Data for $45\leq n\leq 86$

$n$ | ${\mathcal{C}}_{1,n}$ | $\log{\mathcal{D}}_{\text{Chud},n}$ | $\log{\mathcal{D}}_{1,n}$ | $\log{\mathcal{D}}_{2,n}$ | $\log n\mu_{n}$ | $m_{1,\text{max}}$ | $m_{2,\text{max}}$ | $r_{1,\text{max}}$ | $r_{2,\text{max}}$ | $r_{\text{comp}}$ | $N$
---|---|---|---|---|---|---|---|---|---|---|---
87 | $3\cdot 10^{18}$ | 4.574 | 4.632 | 4.830 | 5.136 | 35 | 35 | 1398 | 33 | $38,400,000$ | 14
88 | $10^{23}$ | 4.842 | 4.888 | 5.115 | 5.411 | 9 | 23 | 1097 | 31 | $48,500,000$ | 18
89 | $2\cdot 10^{12}$ | 3.966 | 4.051 | 4.264 | 4.540 | 37 | 35 | 180 | 25 | $25,000,000$ | 9
90 | $5\cdot 10^{25}$ | 5.571 | 5.605 | 5.775 | 6.145 | 23 | 29 | 590 | 54 | $48,500,000$ | 24
91 | $9\cdot 10^{15}$ | 4.488 | 4.561 | 4.762 | 5.049 | 5 | 29 | 385 | 57 | $28,700,000$ | 10
92 | $3\cdot 10^{21}$ | 4.788 | 4.836 | 5.042 | 5.358 | 35 | 31 | 499 | 45 | $48,800,000$ | 16
93 | $10^{17}$ | 4.635 | 4.698 | 4.924 | 5.197 | 37 | 35 | 833 | 27 | $35,200,000$ | 13
94 | $2\cdot 10^{17}$ | 4.770 | 4.821 | 4.961 | 5.321 | 23 | 17 | 2659 | 33 | $45,200,000$ | 17
95 | $3\cdot 10^{11}$ | 4.558 | 4.628 | 4.809 | 5.120 | 41 | 46 | 591 | 42 | $29,900,000$ | 11
96 | $2\cdot 10^{31}$ | 5.239 | 5.281 | 5.462 | 5.807 | 7 | 7 | 1661 | 46 | $47,300,000$ | 19
97 | $5\cdot 10^{14}$ | 4.050 | 4.140 | 4.344 | 4.623 | 45 | 36 | 332 | 17 | $22,900,000$ | 8
98 | $9\cdot 10^{16}$ | 5.038 | 5.085 | 5.339 | 5.603 | 9 | 37 | 375 | 50 | $49,100,000$ | 19
99 | $2\cdot 10^{18}$ | 4.819 | 4.881 | 5.101 | 5.385 | 28 | 32 | 971 | 79 | $34,200,000$ | 14
100 | $2\cdot 10^{23}$ | 5.133 | 5.178 | 5.405 | 5.701 | 23 | 41 | 1587 | 45 | $47,400,000$ | 18

Table 3. Data for $87\leq n\leq 100$ $n$ | ${\mathcal{C}}_{1,n}$ | $\log{\mathcal{D}}_{\text{Chud},n}$ | $\log{\mathcal{D}}_{1,n}$ | $\log{\mathcal{D}}_{2,n}$ | $\log n\mu_{n}$ | $r_{1,\text{max}}$ | $r_{2,\text{max}}$ | $r_{\text{comp}}$ | $N$
---|---|---|---|---|---|---|---|---|---
101 | $3\cdot 10^{8}$ | 4.089 | 4.188 | 4.247 | 4.662 | 253 | 253 | $21,700,000$ | 8
103 | $2\cdot 10^{5}$ | 4.108 | 4.206 | 4.264 | 4.681 | 271 | 37 | $21,900,000$ | 8
107 | $3\cdot 10^{3}$ | 4.145 | 4.249 | 4.323 | 4.717 | 42 | 42 | $21,000,000$ | 8
109 | $8\cdot 10^{3}$ | 4.163 | 4.270 | 4.302 | 4.735 | 147 | 85 | $19,800,000$ | 7
113 | $9\cdot 10^{5}$ | 4.198 | 4.305 | 4.395 | 4.770 | 117 | 33 | $19,400,000$ | 8
127 | $4\cdot 10^{3}$ | 4.311 | 4.428 | 4.507 | 4.883 | 47 | 47 | $17,600,000$ | 6
131 | $2\cdot 10^{9}$ | 4.341 | 4.464 | 4.550 | 4.913 | 193 | 193 | $16,800,000$ | 6
137 | $5\cdot 10^{5}$ | 4.385 | 4.508 | 4.645 | 4.957 | 62 | 62 | $15,700,000$ | 6
139 | $7\cdot 10^{3}$ | 4.399 | 4.527 | 4.551 | 4.971 | 177 | 177 | $15,600,000$ | 6
149 | $224$ | 4.467 | 4.607 | 4.634 | 5.038 | 181 | 19 | $14,900,000$ | 5
151 | $122$ | 4.480 | 4.621 | 4.624 | 5.051 | 71 | 71 | $14,800,000$ | 5
157 | $821$ | 4.518 | 4.657 | 4.687 | 5.089 | 71 | 71 | $14,000,000$ | 5
163 | $10$ | 4.555 | 4.701 | 4.701 | 5.126 | 1 | – | $13,800,000$ | 6
167 | $3\cdot 10^{4}$ | 4.578 | 4.733 | 4.766 | 5.149 | 163 | 163 | $12,800,000$ | 5
173 | $94$ | 4.613 | 4.768 | 4.768 | 5.184 | 253 | – | $13,000,000$ | 5
179 | $15$ | 4.646 | 4.806 | 4.806 | 5.217 | 263 | – | $12,800,000$ | 5
181 | $8\cdot 10^{3}$ | 4.657 | 4.821 | 4.856 | 5.228 | 145 | 23 | $11,800,000$ | 5
191 | $705$ | 4.710 | 4.881 | 4.949 | 5.280 | 29 | 29 | $11,900,000$ | 4
193 | $22$ | 4.720 | 4.895 | 4.895 | 5.291 | 17 | – | $11,900,000$ | 5
197 | $490$ | 4.740 | 4.913 | 4.940 | 5.311 | 61 | 61 | $10,900,000$ | 4
199 | $59$ | 4.750 | 4.930 | 4.930 | 5.321 | 18 | – | $10,900,000$ | 5
211 | $18$ | 4.808 | 4.992 | 4.992 | 5.378 | 25 | – | $10,900,000$ | 4
223 | $205$ | 4.862 | 5.057 | 5.069 | 5.432 | 61 | 61 | $9,800,000$ | 4
227 | $11$ | 4.879 | 5.076 | 5.076 | 5.449 | 1 | – | $9,900,000$ | 4
229 | $28$ | 4.888 | 5.088 | 5.088 | 5.458 | 17 | – | $9,800,000$ | 4
233 | $14$ | 4.905 | 5.108 | 5.108 | 5.475 | 87 | – | $10,000,000$ | 4
239 | $53$ | 4.930 | 5.134 | 5.134 | 5.500 | 33 | – | $9,900,000$ | 4
241 | $2\cdot 10^{3}$ | 4.938 | 5.147 | 5.222 | 5.508 | 31 | 31 | $9,000,000$ | 3
251 | $12$ | 4.978 | 5.193 | 5.193 | 5.548 | 1 | – | $9,000,000$ | 4
257 | $190$ | 5.002 | 5.221 | 5.230 | 5.571 | 73 | 73 | $9,000,000$ | 3
263 | $12$ | 5.024 | 5.241 | 5.241 | 5.594 | 1 | – | $9,000,000$ | 4
269 | $20$ | 5.046 | 5.270 | 5.270 | 5.616 | 7 | – | $9,000,000$ | 3
271 | $72$ | 5.054 | 5.286 | 5.286 | 5.623 | 13 | – | $8,000,000$ | 3
277 | $479$ | 5.075 | 5.314 | 5.336 | 5.645 | 73 | 73 | $8,000,000$ | 3
281 | $36$ | 5.089 | 5.325 | 5.325 | 5.659 | 13 | – | $8,000,000$ | 3
283 | $131$ | 5.096 | 5.334 | 5.340 | 5.666 | 45 | 45 | $7,900,000$ | 3
293 | $12$ | 5.131 | 5.373 | 5.373 | 5.700 | 1 | – | $7,900,000$ | 3
307 | $18$ | 5.177 | 5.426 | 5.426 | 5.746 | 7 | – | $8,000,000$ | 3
311 | $12$ | 5.189 | 5.454 | 5.454 | 5.759 | 1 | – | $7,000,000$ | 3
313 | $7\cdot 10^{3}$ | 5.196 | 5.456 | 5.489 | 5.765 | 129 | 129 | $7,000,000$ | 3
317 | $12$ | 5.208 | 5.473 | 5.473 | 5.778 | 1 | – | $7,000,000$ | 3
331 | $22$ | 5.251 | 5.528 | 5.528 | 5.820 | 9 | – | $7,000,000$ | 3
Table 4. Data for $101\leq n\leq 331$, prime $n$ | ${\mathcal{C}}_{1,n}$ | $\log{\mathcal{D}}_{\text{Chud},n}$ | $\log{\mathcal{D}}_{1,n}$ | $\log{\mathcal{D}}_{2,n}$ | $\log n\mu_{n}$ | $r_{1,\text{max}}$ | $r_{2,\text{max}}$ | $r_{\text{comp}}$ | $N$
---|---|---|---|---|---|---|---|---|---
337 | $13$ | 5.269 | 5.547 | 5.547 | 5.838 | 1 | – | $7,000,000$ | 3
347 | $35$ | 5.298 | 5.579 | 5.579 | 5.867 | 25 | – | $7,000,000$ | 3
349 | $34$ | 5.303 | 5.586 | 5.586 | 5.872 | 25 | – | $7,000,000$ | 3
353 | $13$ | 5.315 | 5.610 | 5.610 | 5.884 | 1 | – | $7,000,000$ | 3
359 | $15$ | 5.331 | 5.629 | 5.629 | 5.900 | 7 | – | $6,000,000$ | 3
367 | $13$ | 5.353 | 5.651 | 5.651 | 5.922 | 1 | – | $6,000,000$ | 3
373 | $31$ | 5.369 | 5.673 | 5.673 | 5.938 | 59 | – | $6,000,000$ | 2
379 | $13$ | 5.385 | 5.691 | 5.691 | 5.954 | 1 | – | $6,000,000$ | 2
383 | $13$ | 5.395 | 5.705 | 5.705 | 5.964 | 1 | – | $6,000,000$ | 2
389 | $13$ | 5.410 | 5.726 | 5.726 | 5.979 | 1 | – | $6,000,000$ | 3
397 | $16$ | 5.431 | 5.753 | 5.753 | 6.000 | 9 | – | $6,000,000$ | 2
401 | $13$ | 5.441 | 5.762 | 5.762 | 6.009 | 65 | – | $6,000,000$ | 2
409 | $15$ | 5.460 | 5.785 | 5.785 | 6.029 | 13 | – | $6,000,000$ | 2
419 | $13$ | 5.484 | 5.815 | 5.815 | 6.053 | 1 | – | $6,000,000$ | 2
421 | $13$ | 5.489 | 5.811 | 5.811 | 6.058 | 1 | – | $6,000,000$ | 2
431 | $22$ | 5.512 | 5.859 | 5.859 | 6.081 | 13 | – | $5,000,000$ | 2
433 | $13$ | 5.516 | 5.867 | 5.867 | 6.085 | 1 | – | $5,000,000$ | 2
439 | $13$ | 5.530 | 5.880 | 5.880 | 6.099 | 1 | – | $5,000,000$ | 2
443 | $13$ | 5.539 | 5.888 | 5.888 | 6.108 | 1 | – | $5,000,000$ | 2
449 | $13$ | 5.552 | 5.905 | 5.905 | 6.121 | 1 | – | $5,000,000$ | 2
457 | $13$ | 5.570 | 5.937 | 5.937 | 6.139 | 1 | – | $5,000,000$ | 2
461 | $13$ | 5.578 | 5.943 | 5.943 | 6.147 | 1 | – | $5,000,000$ | 2
463 | $13$ | 5.583 | 5.953 | 5.953 | 6.152 | 1 | – | $5,000,000$ | 2
467 | $13$ | 5.591 | 5.962 | 5.962 | 6.160 | 1 | – | $5,000,000$ | 2
479 | $14$ | 5.616 | 5.982 | 5.982 | 6.185 | 1 | – | $5,000,000$ | 2
487 | $14$ | 5.633 | 6.014 | 6.014 | 6.201 | 1 | – | $5,000,000$ | 2
491 | $14$ | 5.641 | 6.018 | 6.018 | 6.210 | 1 | – | $5,000,000$ | 2
499 | $30$ | 5.657 | 6.038 | 6.038 | 6.226 | 33 | – | $5,000,000$ | 2
503 | $14$ | 5.665 | 6.048 | 6.048 | 6.233 | 1 | – | $5,000,000$ | 2
509 | $14$ | 5.677 | 6.064 | 6.064 | 6.245 | 1 | – | $5,000,000$ | 2
521 | $14$ | 5.700 | 6.102 | 6.102 | 6.268 | 1 | – | $5,000,000$ | 2
523 | $14$ | 5.703 | 6.099 | 6.099 | 6.272 | 1 | – | $5,000,000$ | 2
541 | $14$ | 5.737 | 6.158 | 6.158 | 6.306 | 1 | – | $4,000,000$ | 2
547 | $14$ | 5.748 | 6.172 | 6.172 | 6.316 | 1 | – | $4,000,000$ | 2
557 | $14$ | 5.766 | 6.202 | 6.202 | 6.334 | 1 | – | $4,000,000$ | 2
563 | $14$ | 5.777 | 6.209 | 6.209 | 6.345 | 1 | – | $4,000,000$ | 2
569 | $14$ | 5.787 | 6.226 | 6.226 | 6.356 | 1 | – | $4,000,000$ | 2
571 | $14$ | 5.791 | 6.246 | 6.246 | 6.359 | 1 | – | $4,000,000$ | 2
577 | $14$ | 5.801 | 6.238 | 6.238 | 6.369 | 1 | – | $4,000,000$ | 2
587 | $14$ | 5.818 | 6.264 | 6.264 | 6.386 | 1 | – | $4,000,000$ | 2
593 | $14$ | 5.828 | 6.290 | 6.290 | 6.396 | 1 | – | $4,000,000$ | 2
599 | $14$ | 5.838 | 6.283 | 6.283 | 6.406 | 1 | – | $4,000,000$ | 2
Table 5. Data for $337\leq n\leq 599$, prime $n$ | ${\mathcal{C}}_{1,n}$ | $\log{\mathcal{D}}_{\text{Chud},n}$ | $\log{\mathcal{D}}_{1,n}$ | $\log{\mathcal{D}}_{2,n}$ | $\log n\mu_{n}$ | $r_{1,\text{max}}$ | $r_{2,\text{max}}$ | $r_{\text{comp}}$ | $N$
---|---|---|---|---|---|---|---|---|---
601 | $14$ | 5.841 | 6.304 | 6.304 | 6.410 | 1 | – | $4,000,000$ | 2
607 | $14$ | 5.851 | 6.323 | 6.323 | 6.420 | 1 | – | $4,000,000$ | 2
613 | $14$ | 5.861 | 6.323 | 6.323 | 6.429 | 1 | – | $4,000,000$ | 1
617 | $14$ | 5.867 | 6.330 | 6.330 | 6.436 | 1 | – | $4,000,000$ | 2
619 | $14$ | 5.871 | 6.328 | 6.328 | 6.439 | 1 | – | $4,000,000$ | 2
631 | $14$ | 5.890 | 6.363 | 6.363 | 6.458 | 1 | – | $4,000,000$ | 2
641 | $14$ | 5.905 | 6.381 | 6.381 | 6.474 | 1 | – | $4,000,000$ | 2
643 | $14$ | 5.908 | 6.391 | 6.391 | 6.477 | 1 | – | $4,000,000$ | 2
647 | $14$ | 5.914 | 6.388 | 6.388 | 6.483 | 1 | – | $4,000,000$ | 1
653 | $14$ | 5.924 | 6.398 | 6.398 | 6.492 | 1 | – | $4,000,000$ | 2
659 | $14$ | 5.933 | 6.410 | 6.410 | 6.501 | 1 | – | $4,000,000$ | 2
661 | $14$ | 5.936 | 6.411 | 6.411 | 6.504 | 1 | – | $4,000,000$ | 2
673 | $14$ | 5.954 | 6.464 | 6.464 | 6.522 | 1 | – | $4,000,000$ | 2
677 | $14$ | 5.959 | 6.449 | 6.449 | 6.528 | 1 | – | $4,000,000$ | 2
683 | $14$ | 5.968 | 6.459 | 6.459 | 6.537 | 1 | – | $4,000,000$ | 2
691 | $14$ | 5.980 | 6.480 | 6.480 | 6.548 | 1 | – | $4,000,000$ | 2
701 | $14$ | 5.994 | 6.498 | 6.498 | 6.562 | 1 | – | $4,000,000$ | 2
709 | $15$ | 6.005 | 6.497 | 6.497 | 6.574 | 1 | – | $4,000,000$ | 2
719 | $14$ | 6.019 | 6.567 | 6.567 | 6.588 | 1 | – | $3,000,000$ | 1
727 | $14$ | 6.030 | 6.585 | 6.585 | 6.599 | 1 | – | $3,000,000$ | 1
733 | $14$ | 6.038 | 6.587 | 6.587 | 6.607 | 1 | – | $3,000,000$ | 1
739 | $14$ | 6.046 | 6.607 | 6.607 | 6.615 | 1 | – | $3,000,000$ | 1
743 | $14$ | 6.052 | 6.613 | 6.613 | 6.620 | 1 | – | $3,000,000$ | 1
751 | $14$ | 6.062 | 6.624 | 6.624 | 6.631 | 1 | – | $3,000,000$ | 1
757 | $14$ | 6.070 | 6.621 | 6.621 | 6.639 | 1 | – | $3,000,000$ | 1
Table 6. Data for $601\leq n\leq 757$, prime

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