\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{amscd} \usepackage{graphicx} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics} \usepackage{latexsym} \usepackage{epsf} \usepackage{breakurl} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{https://oeis.org/#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\Q}{\mathbb Q} \newcommand{\Z}{\mathbb Z} \begin{center} \vskip 1cm{\LARGE\bf Euler's Divergent Series in \\ \vskip .1in Arithmetic Progressions} \vskip 1cm \large Anne-Maria Ernvall-Hyt\"onen\footnote{Supported by the Finnish Cultural Foundation.}\\ Matematik och Statistik\\ {\AA}bo Akademi University\\ Domkyrkotorget 1\\ 20500 {\AA}bo\\ Finland\\ \href{mailto:anne-maria.ernvall-hytonen@abo.fi}{\tt anne-maria.ernvall-hytonen@abo.fi} \\ \ \\ Tapani Matala-aho and Louna Sepp\"al\"a\footnote{Supported by the University of Oulu Scholarship Foundation and the Vilho, Yrj\"o and Kalle V\"ais\"al\"a Foundation.}\\ Matematiikka\\ PL 8000\\ 90014 Oulun yliopisto\\ Finland\\ \href{mailto:tapani.matala-aho@oulu.fi}{\tt tapani.matala-aho@oulu.fi}\\ \href{mailto:louna.seppala@oulu.fi}{\tt louna.seppala@oulu.fi} \end{center} \vskip .2 in \begin{abstract} Let $\xi$ and $m$ be integers satisfying $\xi\ne 0$ and $m\ge 3$. We show that for any given integers $a$ and $b$, $b \neq 0$, there are $\frac{\varphi(m)}{2}$ reduced residue classes modulo $m$ each containing infinitely many primes $p$ such that $a-bF_p(\xi) \ne 0$, where $F_p(\xi)=\sum_{n=0}^\infty n!\xi^n$ is the $p$-adic evaluation of Euler's factorial series at the point $\xi$. \end{abstract} \section{Introduction and results} Euler's factorial series is defined as the sum \begin{equation}\label{hyperseries} F(z):={}_2F_0(1,1\mid z)=\sum_{n=0}^\infty n!z^n. \end{equation} It is clear that in the standard Archimedean metric, it only converges when $z=0$. In the case of the $p$-adic metric, however, the situation changes drastically. For a prime $p$ the normalization $|p|_p=p^{-1}$ gives the usual $p$-adic absolute value. The $p$-adic completion of the rationals $\mathbb{Q}$ with respect to the metric $|\cdot|_p$ is denoted by $\mathbb{Q}_p$. Now the series \eqref{hyperseries} converges in the disc $\left\{z\in\mathbb{Q}_p \; \left| \; |z|_p < p^{\frac{1}{p-1}} \right. \right\}$ and consequently defines a function $F_p$ in that disc by the values $F_p(z):=\sum_{n=0}^\infty n!z^n$. We are interested in arithmetical properties of the values $F_p(\xi)$ of Euler's factorial series at non-zero integer points $\xi \in \Z \setminus \{0\}$. It is an open question whether, e.g., the values $F_p(\pm 1)$ are irrational or even non-zero, which is why it has become customary to study global relations. Let $P(x)\in \mathbb{Z}[x]$, $d := \deg P(x)\ge 1$. For a given $\xi$, a \emph{global relation of degree $d$} is any polynomial identity $P(F_p(\xi))=0$ which is satisfied for all the primes $p$ such that $F_p(\xi)$ is defined. There are several papers considering global relations of the series \eqref{hyperseries} and its generalizations, including generalized hypergeometric series; see, e.g., \cite{BCY04, Chirskii1992, Chirskii2014}. Chirski\u \i \ \cite{Chirskii2015} and Matala-aho and Zudilin \cite{TapaniWadim} have recently investigated first degree global relations of Euler's factorial series. Chirski\u \i \ proved (with our notation) \begin{proposition}\label{Chirskiiglobal}\cite{Chirskii2015} There exist infinitely many primes $p$ such that $F_p(1) \ne 0$. \end{proposition} Matala-aho and Zudilin \cite{TapaniWadim} proved \begin{proposition}\label{TAWA}\cite{TapaniWadim} Given $\xi\in\mathbb{Z}\setminus\{0\}$, let $R\subseteq \mathbb{P}$ be such that \begin{equation}\label{riistajaehto} \limsup_{n\to\infty} c^nn!\prod_{p\in R}|n!|_p^2=0, \quad\text{where}\; c=c(\xi;R):=4|\xi|\prod_{p\in R}|\xi|_p^2. \end{equation} Then either there exists a prime $p\in R$ for which $F_p(\xi)$ is irrational, or there are two distinct primes $p,q\in R$ such that $F_p(\xi)\ne F_q(\xi)$ \textup(while $F_p(\xi), F_q(\xi)\in \mathbb{Q}$\textup). \end{proposition} From now on, let $\Lambda(x)=a-bx\in\mathbb{Z}[x]$, $b\ne 0$. Note that Proposition \ref{Chirskiiglobal} corresponds to the case $\Lambda(x)=x$ and Proposition \ref{TAWA} to the case $\Lambda(x)=a-bx$. Based on Proposition \ref{TAWA}, we are able to prove a more extensive result, Theorem \ref{AB}, which we shall formulate by using the polynomial $\Lambda(x)=a-bx$. For the rest of the work we assume $\xi\in\mathbb{Z}\setminus\{0\}$. \begin{theorem}\label{AB} Let $T\subseteq\mathbb{P}$ be a subset of primes such that the set $T\setminus S$ satisfies condition \eqref{riistajaehto} for any finite subset $S$ of $T$. Then there exist infinitely many primes $p\in T$ such that $\Lambda(F_p(\xi))\ne 0$. \end{theorem} \begin{proof} Define \begin{equation*} \mathcal{R}:=\left\{R\subseteq \mathbb{P} \; | \; R\ \text{satisfies condition}\ \eqref{riistajaehto} \right\}. \end{equation*} If $R\in \mathcal{R}$, the proof of Proposition \ref{TAWA} in \cite{TapaniWadim} shows that there exists at least one prime $p\in R$ such that $\Lambda(F_p(\xi)) \ne 0$. Take now a subset $T$ of primes such that $T\setminus S \in \mathcal{R}$ for any finite subset $S$ of $T$. Define a new set \begin{equation*} A:= \left\{p\in T\; |\; \Lambda(F_p(\xi)) \ne 0 \right\}. \end{equation*} If the set $A$ is finite, then $T\setminus A\in \mathcal{R}$ by assumption. But \begin{equation*} T\setminus A=\{p\in T\; |\; \Lambda(F_p(\xi))= 0 \}. \end{equation*} This is a contradiction, and thus $\# A=\infty$. \end{proof} From Theorem \ref{AB} it follows that for any set $R\in \mathcal{R}$ satisfying the assumptions of Theorem \ref{AB}, there exist infinitely many primes $p\in R$ such that $F_p(\xi) \ne \frac{a}{b}$. In particular, if we take $T=\mathbb{P}$, then we see immediately that any $\mathbb{P}\setminus S \in \mathcal{R}$, if $S$ is a finite set of primes. Thus, Theorem \ref{AB} implies the following corollary. \begin{corollary}\label{Corkaikkip} Let $\frac{a}{b} \in \mathbb{Q}$ be given. Then there exist infinitely many primes $p\in \mathbb{P}$ such that $F_p(\xi) \ne \frac{a}{b}$. \end{corollary} This still seems to be far from implying irrationality, for the prime $p=p_{\Lambda}\in R$ for which $F_p(\xi) \ne \frac{a}{b}$, may depend on the polynomial $\Lambda(x)=a-bx$. We note that there are results (see, e.g., \cite{BCY04, Chirskii1992}) from which Corollary \ref{Corkaikkip} follows, but the proof presented here is different from these earlier articles. As will be seen shortly, we may also considerably diminish the prime number set where Corollary \ref{Corkaikkip} is still valid. The question rises whether, for example, the reduced residue system modulo $m$, $m\in\mathbb{Z}_{\ge 3}$, could produce examples of prime subsets satisfying condition \eqref{riistajaehto}. Indeed, that is the case, as will be demonstrated in the following theorem, the main result of this paper. Let $m \in \Z_{\ge 3}$ and denote $\overline{h} := \{ h+km \; | \; k \in \Z \}$. We write $\overline{h}_1, \ldots, \overline{h}_{\varphi (m)}$ for the $\varphi (m)$ residue classes in the reduced residue system modulo $m$. Dirichlet's theorem about primes in arithmetic progressions says that each of these classes contains infinitely many prime numbers. \begin{theorem}\label{epatasa} Let $m\in\mathbb{Z}_{\ge 3}$ be a given integer. Assume that $R=\bigcup_{j=1}^r \left( \overline{h}_{i_j} \cap \mathbb{P} \right)$ is any union of the primes in $r$ residue classes in the reduced residue system modulo $m$, where $r > \frac{\varphi(m)}{2}$. Then there are infinitely many primes $p \in R$ such that $\Lambda(F_p(\xi))\ne 0$. \end{theorem} Observe that the `relation set' $G_{\Lambda}:=\{p\in\mathbb{P}\;|\; \Lambda(F_p(\xi))=0\}$ cannot be too big: By Theorem \ref{AB}, the set $G_{\Lambda}$ obviously cannot satisfy condition \eqref{riistajaehto}. Theorem \ref{epatasa} shows that the set $G_\Lambda$ cannot contain the primes of more than half of the reduced residue classes modulo $m$. Therefore $F_p(\xi) \ne \frac{a}{b}$ holds---in the above sense---for at least `half' of all the primes $p\in\mathbb{P}$. Theorem \ref{epatasa} implies that there is a residue class modulo $m$ containing infinitely many primes $p$ for which $\Lambda(F_p(\xi))\ne 0$. Since the union $R$ may be chosen arbitrarily, we actually obtain: \begin{corollary}\label{corollary6} Let $m\in\mathbb{Z}_{\ge 3}$ be a given integer. There are $\frac{\varphi(m)}{2}$ reduced residue classes modulo $m$ each containing infinitely many primes $p$ such that $F_p(\xi) \ne \frac{a}{b}$. \end{corollary} \begin{proof} By Theorem \ref{epatasa}, the union $\bigcup_{i=1}^{\frac{\varphi(m)}{2}+1} \overline{h}_i$ contains infinitely many primes $p$ such that $\Lambda(F_p(\xi))\ne 0$. Thus one of the reduced residue classes $\overline{h}_1, \ldots, \overline{h}_{\frac{\varphi(m)}{2}+1}$ must contain infinitely many such primes---suppose it is $\overline{h}_1$. Now we may apply Theorem \ref{epatasa} again to the union $\bigcup_{i=2}^{\frac{\varphi(m)}{2}+2} \overline{h}_i$, and without loss of generality assume that this time the residue class containing infinitely many of those certain primes is $\overline{h}_2$. This procedure can be repeated $\frac{\varphi (m)}{2}$ times, from which the assertion follows. \end{proof} The number $\frac{\varphi(m)}{2}$ is always an integer because $\varphi(m)$ is even when $m\ge 3$. The case $m=2$ is of no interest because all the primes except the prime $2$ are in the same residue class. Let $P(x) \in \Z[x]$ be a polynomial of degree $d$ and write it in the form \begin{align*} P(x) = \;&a_d (x+1)(x+2) \cdots (x+d) + a_{d-1} (x+1)(x+2) \cdots (x+d-1) \\ &+ \cdots + a_2 (x+1)(x+2) + a_1(x+1) + a_0, \end{align*} where $a_i \in \Z$, $i=0,1,\ldots,d$. Chirski\u\i \ \cite{Chirskii2017} has shown that in any $\Q_p$, $$ \sum_{n=0}^\infty P(n) n! = A F_p(1) + B, $$ where $$ A:= \sum_{i=0}^d a_i \in \Z, \quad B:= - \left( a_d \sum_{n=0}^{d-1} n! + a_{d-1} \sum_{n=0}^{d-2} n! + \cdots + a_2 1! + a_1 \right) \in \Z. $$ Thus we may state another corollary: \begin{corollary}\label{corollary7} Let $m\in\mathbb{Z}_{\ge 3}$ be a given integer and $P(x) \in \Z[x]$ a polynomial such that $A \neq 0$ in the above notation. Then, for any $s, t \in \Z$, $t \neq 0$, there are $\frac{\varphi(m)}{2}$ reduced residue classes modulo $m$ each containing infinitely many primes $p$ such that $$ \sum_{n=0}^\infty P(n) n! \ne \frac{s}{t}. $$ \end{corollary} \begin{proof} Choose $\xi=1$, $a= s-tB$, and $b=tA \neq 0$ in Corollary \ref{corollary6}. It follows that there are $\frac{\varphi(m)}{2}$ reduced residue classes modulo $m$ each containing infinitely many primes $p$ such that $$ s-t\sum_{n=0}^\infty P(n) n! = s-tB - tAF_p(1) = a-bF_p(1) \neq 0. $$ \end{proof} Finally, assuming the generalized Riemann hypothesis (GRH), we can say something slightly different, namely that in any collection of $\frac{\varphi(m)}{2}$ residue classes in the reduced residue system modulo $m$, there is at least one prime satisfying the condition $\Lambda(F_p(\xi))\ne 0$ but now for a $\xi$ satisfying the condition of Theorem \ref{tasa}. The previous theorem gave us the existence of infinitely many such primes in some $\frac{\varphi(m)}{2}$ residue classes. Now we can prove that also in the complement of the previous $\frac{\varphi(m)}{2}$ residue classes, there must be at least one prime satisfying the condition. \begin{theorem}\label{tasa} Assume the GRH. Let $m\in\mathbb{Z}_{\ge 3}$ be a given integer. Assume that $R=\bigcup_{j=1}^{\varphi(m)/2} \left( \overline{h}_{i_j} \cap \mathbb{P} \right)$ is any union of the primes in $\frac{\varphi(m)}{2}$ residue classes in the reduced residue system modulo $m$. Then there is a value $d_m$ such that if $\xi$ is any non-zero integer satisfying the bound \[ 4|\xi|\prod_{p\in R}|\xi|_p^2 \frac{\varphi(m)}{2}$. It suffices to prove that for any non-zero integer $\xi$ and a finite subset $S=\{p_1,\ldots,p_k\} \subseteq R$, the condition \[ \lim\sup_{n\rightarrow \infty} c_0^{n}n! \prod_{p\in R\setminus S}|n!|_p^2=0, \quad c_0=4|\xi|\prod_{p\in R\setminus S}|\xi|_p^2, \] is satisfied. Thus we are led to study the expression \begin{equation}\label{logeq} \log \left(c_0^{n}n! \prod_{p\in R \setminus S}|n!|_p^2\right)=n\log c_0+\log n!+2\sum_{p\in R \setminus S} \log |n!|_p. \end{equation} Here \begin{equation}\label{Ssumma} 2\sum_{p\in S}\log |n!|_p=2\sum_{i=1}^k\log |n!|_{p_i}=O(n) \end{equation} when $n$ grows for any given fixed $S$. The constant implied by the $O$-term may be arbitrarily large, but it is constant in the $n$ aspect. Recall the Stirling formula (see, e.g., \cite[formula 6.1.38]{stirling}): \[ \log n!=\log \sqrt{2\pi}+\left(n+\frac{1}{2}\right)\log n-n+\frac{\theta(n)}{12}, \] where $0<\theta(n)<1$. The above can be further simplified to $\log n!=n\log n+O(n)$. Combining this with \eqref{logeq}, \eqref{Ssumma}, and Lemma \ref{lemma2}, we get \begin{align*} \log \left(c_0^{n}n! \prod_{p\in R\setminus S}|n!|_p^2\right)&=n\log n+O(n)+2\sum_{j=1}^r \sum_{p \in \overline{h}_{i_j}} \log |n!|_p - 2 \sum_{i=1}^k \log |n!|_{p_i}\\ &=n\log n+O(n)-\frac{2rn\log n}{\varphi(m)}+O(n\log \log n)\\ &=n\log n\left(1-\frac{2r}{\varphi(m)}\right)+O(n)+O(n\log \log n) \rightarrow -\infty \end{align*} as $n \to \infty$ because the coefficient $1-\frac{2r}{\varphi(n)}$ of the main term is negative. The result follows from Theorem \ref{AB}. \end{proof} Let us now move to the proof of Theorem \ref{tasa}. \begin{proof}[Proof of Theorem \ref{tasa}] Here we use Proposition \ref{TAWA}, so we need to check condition \eqref{riistajaehto} with $c=4|\xi|\prod_{p\in R}|\xi|_p^2$. When looking at the terms in \[ \log \left(c^{n}n! \prod_{p\in R}|n!|_p^2\right)=n\log c+\log n!+2\sum_{p\in R} \log |n!|_p, \] we again use Stirling's formula and the bound for $|n!|_p$ as earlier. Now the main terms cancel, so what is left is $O(n)$ for some constant depending only on $m$. Therefore, the contribution of this term can be cancelled if $c$ is sufficiently small, namely, below some $d_m$ depending only on $m$. \end{proof} \begin{remark} This proof of Theorem \ref{tasa} cannot be generalized to the case with infinitely many primes because the constant implied by the $O$-term in contribution of the arbitrary subset $S$ can be arbitrarily large, and therefore, we cannot use the argument of a term of magnitude $n$ cancelling the other terms. \end{remark} \section{Acknowledgments} We wish to thank the referee for their careful reading of the manuscript and the valuable suggestion of adding Corollary \ref{corollary7}. \begin{thebibliography}{99} \bibitem{stirling} M. 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Number Theory} \textbf{179} (2017), 220--239. \bibitem{TapaniWadim} T. Matala-aho and W. Zudilin, Euler's factorial series and global relations, \emph{J. Number Theory} \textbf{186} (2018), 202--210. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 11J61. \noindent \emph{Keywords:} divergent series, global relation, $p$-adic number. \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received November 7 2018; revised version received January 14 2019. Published in {\it Journal of Integer Sequences}, February 22 2019. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}