\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}{\rm \underline{#1}}} \def\modd#1 #2{#1\ \mbox{\rm (mod}\ #2\mbox{\rm )}} \newcommand{\mean}{\mathop{\mathrm{mean}}} \newcommand{\lcm}{\mathop{\mathrm{lcm}}} \newcommand{\li}{\mathop{\mathrm{li}}} \newcommand{\Exp}{\mathop{\mathrm{Exp}}} \newcommand{\Gumbel}{\mathop{\mathrm{Gumbel}}} \newcommand{\itpara}[1]{\smallskip{\it #1}} \newcommand{\bfpara}[1]{\smallskip{\bf #1}} \newcommand{\rmpara}[1]{\smallskip{\rm #1}} \def \BB{{\cal B}} \def \TT{{\cal T}} \def \II{{\cal I}} \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} \begin{center} \vskip 1cm{\LARGE\bf On the First Occurrences of Gaps \\ \vskip 1mm Between Primes in a Residue Class } \vskip 0.6cm {\large Alexei Kourbatov\\ JavaScripter.net \\ Redmond, WA 98052\\ USA \\ \href{mailto:akourbatov@gmail.com}{\tt akourbatov@gmail.com} \\ \ \\ Marek Wolf\\ Faculty of Mathematics and Natural Sciences\\ Cardinal Stefan Wyszy\'nski University \\ Warsaw, PL-01-938 \\ Poland \\ \href{mailto:m.wolf@uksw.edu.pl}{\tt m.wolf@uksw.edu.pl} \\ } \vspace{8mm} {\normalsize\it To the memory of Professor Thomas R.~Nicely (1943--2019)} \vspace{6mm} \end{center} \begin{abstract} We study the first occurrences of gaps between primes in the arithmetic progression (P):~$r$, $r+q$, $r+2q$, $r+3q, \ldots, $ where $q$ and $r$ are coprime integers, $q>r\ge1$. The~growth trend and distribution of the first-occurrence gap sizes are similar to those of {\it maximal} gaps between primes in (P). The histograms of first-occurrence gap sizes, after appropriate rescaling, are well approximated by the Gumbel extreme value distribution. Computations suggest that first-occurrence gaps are much more numerous than maximal gaps: there are $O(\log^2 x)$ first-occurrence gaps between primes in (P) below $x$, while the number of maximal gaps is only $O(\log x)$. We explore the connection between the asymptotic density of gaps of a given size and the corresponding generalization of Brun's constant. For the first occurrence of gap $d$ in (P), we expect the end-of-gap prime $p\asymp\sqrt{d}\exp({\sqrt{d/\varphi(q)}})$ infinitely often. Finally, we study the gap size as a function of its index in the sequence of first-occurrence gaps. \end{abstract} \section{Introduction} Let $p_n$ be the $n$-th prime number, and consider the difference between successive primes, called a {\it prime gap}: $p_{n+1}-p_n$. If a prime gap is {\it larger} than all gaps before it, we call this gap {\it maximal}. If a prime gap is either {\it larger} or {\it smaller} than (but not equal to) any gap before it, we call this gap a {\it first occurrence} \cite{Brent1973,Nicely1999,NicelyNyman,OliveiraeSilva2014,YoungPotler}. Let $G(x)$ be the largest gap between primes $\le x$: $$G(x) = \max_{p_{n+1}\le x} (p_{n+1} - p_n). $$ A lot has been conjectured about large prime gaps; much less has been rigorously proved. Today's best known results on the growth of $G(x)$ as $x\to\infty$ are as follows: $$ A\,\textstyle{\log x \log\log x \log\log\log\log x \over \log\log\log x} < G(x) < B x^{0.525}, $$ with some positive constants $A$ and $B$. The upper bound is due to Baker, Harman, and Pintz~\cite{BakerHarmanPintz}, and the lower bound to Ford, Green, Konyagin, Maynard, and Tao \cite{FGKMT}. Computations suggest that the above double inequality holds with $A=B=1$ for all $x\ge153$, while the lower bound holds with $A=1$ wherever the left-hand side expression is defined; cf.\,\cite{FunkhouserGoldstonLedoan}. The notion of prime gap admits a natural generalization. Given an increasing integer sequence (S), we can consider gaps $p'-p$ between primes $p$ and $p'$ both of which are in (S), under the condition that $p0$ \\ $f(x)\lesssim h(x)$ & $f(x)\sim h(x)$ and $f(x)0; $$ that is, the numbers $\pi(x)$ and $\lfloor \li x \rfloor$ almost agree in the left half of their digits. Let $q>r\ge1$ be fixed coprime integers, and consider the arithmetic progression $$\mbox{(P)} ~~~ r, \ r+q, \ r+2q, \ r+3q, \ \ldots \,.$$ {\it Dirichlet's theorem on arithmetic progressions} \cite{Dirichlet} establishes that there are infinitely many primes in progression (P) whenever $\gcd(q,r)=1$. {\it The prime number theorem for arithmetic progressions} \cite{deKoninckLuca} states that the number of primes in progression (P) not exceeding $x$ is asymptotically equal to $1/\varphi(q)$ of the total number of primes $\le x$, that is, \begin{equation}\label{eq-pntap} \pi(x;q,r) \sim {\pi(x) \over \varphi(q)} \sim {\li x \over \varphi(q)} \qquad\mbox{ as } x\to\infty, \end{equation} where $\varphi(q)$ is Euler's totient function. Under the {\it Generalized Riemann Hypothesis} \cite[p.\,253]{deKoninckLuca} we have $$\pi(x;q,r)-{\li x \over \varphi(q)} \,=\, O\!\left(x^{1/2+\varepsilon}\right) \quad\mbox{ for any } \varepsilon>0. $$ {\it Polignac's conjecture} \cite{polignac1849} states that every gap of an even size $d=2n$, $n\in{\mathbb N}$, actually occurs between consecutive primes infinitely often. (In Section~\ref{sectWhen} we generalize Polignac's conjecture to gaps of size $n\cdot\lcm(2,q)$ between primes in progression (P); see Conjecture \ref{genPolignac}.) Regarding the behavior of the function $G(x)$---the maximal prime gap up to $x$---Cram\'er conjectured in the 1930s that $G(x)=O(\log^2 x)$ \cite{Cramer}; \ Shanks set forth a stronger hypothesis: $G(x)\sim\log^2 x$ \cite{Shanks1964}. Recent models \cite{BanksFordTao,Granville} cast doubt on the latter conjecture; we still consider it plausible in an ``almost all'' sense (for {\it almost all} maximal gaps). Let $G_{q,r}(x)$ denote the maximal gap between primes $\le x$ in progression (P). The \itpara{Generalized Cram\'er conjecture} states that almost all maximal gaps $G_{q,r}(x)$ satisfy \begin{equation}\label{genCramer} G_{q,r}(x) < \varphi(q) \log^2 x. \qquad\mbox{ \cite[eq.~(34)]{KourbatovWolf2019} } \end{equation} The \itpara{Generalized Shanks conjecture} is similar: almost all maximal gaps $G_{q,r}(x)$ satisfy \begin{equation}\label{genShanks} G_{q,r}(x) \sim \varphi(q) \log^2 x. \qquad\mbox{ \cite[eq.~(35)]{KourbatovWolf2019}} \end{equation} \section{The growth trend of large gaps}\label{sec-trend} In this section we introduce three trend functions useful in describing the growth and distribution of large gaps between primes in arithmetic progression (P): $r+nq$. These trend functions are asymptotically equivalent to $\varphi(q)\log^2 x$ and differ only in lower-order terms. \begin{itemize} \item The function $T_0(q,x)$ is called a {\it baseline trend}. Roughly speaking, for large enough $x$, this trend curve separates typical sizes of maximal gaps from typical sizes of non-maximal first-occurrence gaps; see sect.\,\ref{subsec-T0}. \item The function $T_m(q,x)$ will give us an estimate of the most probable maximal gap sizes. We have $T_m(q,x)\ge T_0(q,x)$; see sect.\,\ref{subsec-Tm}. \item The function $T_f(q,x)$ will give us an estimate of the most probable first-occurrence gap sizes. For large $x$, we have $T_f(q,x)1$. \end{definition} Formula (\ref{T0qx}) is a concise form of \cite[eq.\,(33)]{KourbatovWolf2019} in which we set a constant term\footnote{ The choice of $b=\log\varphi(q)$ for use in \cite[eq.\,(33)]{KourbatovWolf2019} reflects our heuristic expectation that, for large $x$, maximal gaps between primes $\le x$ in (P) should depend primarily on the quantities $\varphi(q)$, $a(q,x)$, and $\pi(x)$. Moreover, this choice is supported by extensive numerical results; cf.\,\cite[sect.\,3.1]{KourbatovWolf2019}. } $b=\log\varphi(q)$. Thus, by definition we have \begin{equation}\label{T0qx2logli} T_0(q,x) = {x \varphi(q) \over \li x} \left(2 \log{ \li x \over \varphi(q)} - \log{x\over \varphi(q)} \right). \end{equation} (Equations (\ref{T0qx}) and (\ref{T0qx2logli}) can also be viewed as a generalization of the trend equation derived in \cite{Wolf2011, Wolf2014} for maximal prime gaps $G(x)$, that is, for the particular case $q=2$.) \begin{remark} The baseline trend $T_0$ defined in (\ref{T0qx}) has the following properties: \begin{itemize} \item[(i)] $T_0(q,x)$ is a continuous and smooth function of $x$ for every fixed $q$. \item[(ii)] $T_0(q,x)$ does not depend on the choice of $r$ in progression (P). \item[(iii)] $T_0(q,x)$ does not use unknown parameters determined a posteriori. \end{itemize} \end{remark} By way of comparison, for {\it exponentially distributed random} (i.i.d.)~intervals between rare events, with mean interval $\alpha$ and cdf $\Exp(\xi;\alpha)=1-e^{-\xi/\alpha}$, the most probable maximal interval in $[0,x]$ is about $\alpha \log{x\over \alpha}$. Note the absence of $\li x$, in contrast to (\ref{T0qx}); cf.\,\cite[sect.\,4.1.1]{Gumbel}, \cite[sect.\,8]{Kourbatov2013}, and \cite[sect.\,3.3]{Kourbatov2016}. Clearly, as $x\to\infty$ we have \begin{equation} a(q,x) \lesssim \varphi(q)\log x, \end{equation} \begin{equation} T_0(q,x) \lesssim \varphi(q)\log^2 x. \end{equation} We numerically experimented with arithmetic progressions (P): $r+nq$\, for many different coprime pairs $(q,r)$, up to $q=10^5$. Invariably, we found that the majority of maximal gaps between primes $\le x$ in (P) have sizes {\it above} the baseline trend curve $T_0(q,x)$; that is, we usually have $G_{q,r}(x)>T_0(q,x)$. At the same time, for large enough $x$, a significant proportion of non-maximal first-occurrence gap sizes are {\it below} the curve $T_0(q,x)$. These findings hold at least for $x\le10^{14}$. \subsection{The trend $T_m(q,x)$ of maximal gaps}\label{subsec-Tm} Most probable {\it maximal} gaps $G_{q,r}(x)$ are close to the following trend \cite[eqs.\,(33),\,(45)]{KourbatovWolf2019}: \begin{equation}\label{Tmqx} T_m(q,x) = T_0(q,x) + {\mathcal E}_m(q,x), \end{equation} where the error term ${\mathcal E}_m(q,x)$ is nonnegative and given by this empirical formula: \begin{equation}\label{Emqx} {\mathcal E}_m(q,x) \approx {b_1 \log\varphi(q) \over (\log\log x)^{b_2}}\cdot a(q,x). \end{equation} For $q=2$, formula (\ref{Emqx}) gives ${\mathcal E}_m(2,x)=0$. Indeed, we found that the baseline trend $T_0(2,x)$, even {\it without} an error term, satisfactorily describes the most probable sizes of maximal prime gaps $G(x)$. (Another reasonably good choice of the error term in (\ref{Tmqx}) for $q=2$ is ${\mathcal E}_m(2,x)=ca(q,x)$ with a constant $c\in[0,{1\over2}]$.) For $q>2$, we found that the error term ${\mathcal E}_m(q,x)$ is positive, and it is best to determine $b_1$ and $b_2$ in (\ref{Emqx}) {\it a posteriori}. However, for $x\in[10^7,10^{14}]$ and $q\in[10^2,10^5]$, choosing \begin{equation}\label{b1b2} b_1=4, \qquad b_2=2.7 \end{equation} yields a ``good enough'' trend curve for typical maximal gap sizes. This choice of $b_1$, $b_2$ works best when $q$ is prime or semiprime. See \cite{Kourbatov2016,KourbatovWolf2019} for further details on maximal gaps. \subsection{The trend $T_f(q,x)$ of first-occurrence gaps}\label{subsec-Tf} For the trend equation to better describe the most probable {\it first-occurrence} gap sizes, we add a lower-order correction term to the baseline trend (\ref{T0qx}): \begin{equation}\label{Tfqx} T_f(q,x) = T_0(q,x) + {\mathcal E}_f(q,x). \\ \end{equation} Based on extensive numerical data, we find that the term ${\mathcal E}_f(q,x)$ is eventually negative and given by this empirical formula: \begin{equation}\label{Efqx} {\mathcal E}_f(q,x) \approx (c_0 - c_1 \log \log x)\cdot a(q,x), \end{equation} where $a(q,x)$ is defined by (\ref{aqx}). Like with $T_m(q,x)$, it is best to determine the parameters $c_0$ and $c_1$ in (\ref{Efqx}) {\it a posteriori}. However, in some special cases, we were able to find predictive formulas for $c_0$ and $c_1$ as functions of $q$: \begin{itemize} \item If $q$ is a {\it prime} $\ge5$, the following empirical formulas \begin{equation}\label{c0c1primeq} \begin{split} c_0 &= c_0(q) \approx 2.18 \, (\log\lcm(2,q))^{0.58}, \\ c_1 &= c_1(q) \approx 1.18 \, (\log\lcm(2,q))^{0.364} \end{split} \end{equation} work well for parameters $c_0$ and $c_1$ in the term ${\mathcal E}_f(q,x)$ (\ref{Efqx}), for $x\in[10^7,10^{14}]$ and $q\in[5,10^5]$. \item If $q$ is an {\it even semiprime} $\ge10$, it is plain that we deal with the same sequences of even first-occurrence gaps as above, so we can use the same formulas (\ref{c0c1primeq}). \item For $q=2$ (gaps in the sequence of all primes), the parameters \begin{equation} c_0\approx3, \qquad c_1\approx1.58 \end{equation} are close to optimal in the term ${\mathcal E}_f(2,x)$, at least for $x<2^{64}$. \end{itemize} \subsection{Numerical results} Using {\tt PARI/GP} \cite{Pari} we performed computational experiments with progressions (P): $r+nq\,$ for many different values of $q\in[2,10^5]$. Figure \ref{figq211trend} shows the sizes of first-occurrence gaps between primes in twenty progressions (P), for $q=211$ and $r\in[1,20]$. Results for other values of $q$ look similar to Fig.\,\ref{figq211trend}. All our numerical results show that trend curves of the form (\ref{Tfqx}), (\ref{Efqx}) satisfactorily describe the growth of typical first-occurrence gaps between primes in progressions (P). $$~$$ \begin{figure}[H] \centering \includegraphics[bb=30 0 540 330,width=5.6in,height=4in,keepaspectratio]{akmw2020-q211trendx} \caption{First-occurrence gaps $d_{q,r}$ between primes $p=r+nq\le10^{12}$ for $q=211$, $r\in[1,20]$. Dotted curve $(\bullet)$: trend $T_f$ (\ref{Tfqx})--(\ref{c0c1primeq}); \ blue dotted curve~$(^{_{\color{blue}\,\blacklozenge}})$: baseline trend $T_0$ (\ref{T0qx}); \ top line: $y = \varphi(q) \log^2 p$. The vast majority of {\it maximal} gaps stay above the curve $T_0$, while a growing proportion of non-maximal first-occurrence gaps are observed below $T_0$. } \label{figq211trend} \vspace*{\floatsep} \begin{center} \includegraphics[bb=-6 0 800 180,width=8.65in,height=2in,keepaspectratio]{akmw2020-q211distx} \caption{Histogram of rescaled values $u$ (\ref{rescaledu}) for the first-occurrence gaps between primes $p=r+nq\in[10^7,10^{12}]$ for $q=211$ and all $r\in[1,210]$. Bins:~53,\, bin size $\approx0.23$. Smooth curve: best-fit Gumbel distribution (pdf) with scale $\alpha=0.87692$ and mode $\mu=-0.00902$. } \label{figq211dist} \end{center} \end{figure} \section{The distribution of first-occurrence gap sizes}\label{sect-distribution} Extensive numerical data allows us to observe that typically a first-occurrence gap is also the largest gap between primes in the interval $[x,x+x/(\log x)^{1+\varepsilon}]$, for some $\varepsilon>0$. This invites the question (cf.\,\cite{Kourbatov2013}--\cite{LiPrattShakan2017}): do first-occurrence gap sizes, after an appropriate rescaling transformation, obey the same distribution as maximal gap sizes---the Gumbel extreme value distribution? Let $d=d_{q,r}(x)$ denote the size of the first-occurrence gap between primes $p\in[1,x]\cap$(P); specifically, the gap $d_{q,r}(x)$ refers to the first occurrence {\it closest to} $x$ on the interval $[1,x]$. Infinitely many distinct values of $d_{q,r}(x)$ are maximal gaps, for which $d_{q,r}(x)=G_{q,r}(x)$. At the same time, infinitely many distinct values of $d_{q,r}(x)$ are {\it not} maximal gaps, and for these first occurrences we have $d_{q,r}(x)0$ such that a positive proportion of maximal gaps satisfy the double inequality \begin{equation}\label{dblineq-C0C1} C_0\sqrt{d}\,e^{\sqrt{d/\varphi(q)}} < P_f(d;q,r) < C_1\sqrt{d}\,e^{\sqrt{d/\varphi(q)}}. \end{equation} \end{conjecture} For primes up to $10^{14}$ and $q\le10^5$, computations suggest that if we take $C_0={1\over10}$, $C_1=10$, then the majority of maximal gap sizes will satisfy (\ref{dblineq-C0C1}); a sizable proportion of non-maximal first-occurrence gaps will also be found within the bounds (\ref{dblineq-C0C1}). How did we arrive at the estimates (\ref{dblineq-Pfdqr})--(\ref{dblineq-C0C1})? The left-hand side $e^{\sqrt{d/\varphi(q)}}$ in (\ref{dblineq-Pfdqr}) stems from the conjecture \cite[eq.\,(34)]{KourbatovWolf2019} that {\it almost all} maximal gaps between primes in (P) satisfy $$G_{q,r}(p) < \varphi(q) \log^2p.$$ (This is our generalization of Cram\'er's conjecture \cite{Cramer}.) To see why we have $C\sqrt{d}\,e^{\sqrt{d/\varphi(q)}}$ in the right-hand side of (\ref{dblineq-Pfdqr})--(\ref{dblineq-C0C1}), consider the function \begin{equation}\label{tildePdq} \tilde{P}(d,q) = \sqrt{d}\,e^{\sqrt{d/\varphi(q)}}. \end{equation} For any fixed $q\ge2$, one can check that \begin{equation}\label{limittildePdq} \lim_{x\to\infty}{\tilde{P}(T_0(q,x),q) \over x} = e^{-1/2} \approx 0.60653. \end{equation} That is, for large $d$ and $x$, the function $e^{1/2}\tilde{P}(d,q)$ is ``almost inverse'' of $T_0(q,x)$ in (\ref{T0qx}). \begin{remark} We do not know whether the lower bound in \eqref{dblineq-Pfdqr} is reversed infinitely often or finitely often. For the special case $q=2$ (gaps in the sequence of all primes), heuristic models \cite{BanksFordTao,Granville} lead to $\limsup\limits_{x\to\infty}{G(x)\over\log^2 x}\ge 2e^{-\gamma}\approx1.1229$---suggesting that, for first-occurrence prime gaps, the lower bound in \eqref{dblineq-Pfdqr} is reversed infinitely often. However, it is quite difficult to find examples of progressions (P): $r + nq$ with exceptionally early first occurrences of gaps $d$ such that $P'_f(d;q,r)1$. We know only a modest number of maximal gaps of this kind \cite[Table 1]{KourbatovWolf2019}, \cite{Raab2020}--\cite{Raab2020may}. We have never seen examples of sign reversal of the lower bound in \eqref{dblineq-Pfdqr} ($\mbox{CSG}>1$) for {\it non-maximal} first-occurrence gaps in (P). Note also that Sun \cite[conj.\,2.3]{Sun2013} proposed a hypothesis similar to Firoozbakht's conjecture for gaps between primes in (P), which would imply (at most) a finite number of such sign reversals for any given progression (P). \end{remark} \section{A detour: generalization of Brun's constants}\label{SectBrunConst} We will now give another (less direct) heuristic way of deriving the estimate $C\sqrt{d}\,e^{\sqrt{d/\varphi(q)}}$ in (\ref{dblineq-Pfdqr})--(\ref{dblineq-C0C1}). Our approach is to estimate the sum of reciprocals of prime pairs $(p,\,p+d)$ separated by a given fixed gap $d$. (Wolf \cite{Wolf1997} used this heuristic approach to treat the case $q=2$, obtaining the estimate\footnote{ The preprint \cite{Wolf1997} employs the ``physical'' notation where the relation $f\sim g$ is a shorthand for ``the functions $f$ and $g$ have the same order of magnitude.'' Thus $\sqrt{d}\,e^{\sqrt{d}}$ is just an order-of-magnitude estimate. } $\sqrt{d}\,e^{\sqrt{d}}$ for the first occurrence of gap $d$.) First, let us recall some definitions. {\it Twin primes} are pairs of prime numbers $(p,\,p+2)$ separated by gap 2---the smallest possible gap between odd primes. {\it Cousin primes} are pairs of prime numbers $(p,\,p+4)$ separated by gap 4. Brun \cite{Brun} proved in 1919 that the series consisting of reciprocal twin primes converges. {\it Brun's constant} $B_2$ for twin primes is the sum of reciprocals \cite[\seqnum{A065421}]{Fry_Nesheiwat_Szymanski,Nicely_Brun,Nicely2001,SebahGourdon} $$ B_2 = \left({1\over3}+{1\over5}\right) + \left({1\over5}+{1\over7}\right) + \left({1\over11}+{1\over13}\right) + \cdots \approx 1.90216. $$ A generalization of Brun's theorem exists for series of reciprocal values of prime pairs $(p,\,p+d)$ separated by {\it any fixed gap} $d$ \cite{Segal}. For example, Brun's constant $B_4$ for cousin primes (with the pair $(3,7)$ omitted) is \cite[\seqnum{A194098}]{Wolf1996} $$ B_4 = \left({1\over7}+{1\over11}\right) + \left({1\over13}+{1\over17}\right) + \left({1\over19}+{1\over23}\right) + \cdots \approx 1.19704. $$ Note that twin primes $p$, $p+2$ are always consecutive. Except for the pair $(3,7)$, primes $p$, $p+4$ are also necessarily consecutive. If $d\ge6$, however, there may also be other primes between the primes $p$ and $p+d$. In what follows, we will consider gaps $d\ge6$. Suppose that primes $p$ and $p'$ are in the residue class $r$ (mod $q$), $p2}\frac{p-1}{p-2} \cdot \frac{ \pi^2(x)}{\varphi ^2(q)x} \, e^{-{d\cdot\pi(x)\over\varphi(q)x}}, \end{equation} cf.\,\cite[eq.~(2)]{GoldstonLedoan2011} and \cite[eq.~(30)]{KourbatovWolf2019}. In formula (\ref{main2}), $p$ runs through odd prime factors of $d$, \begin{equation}\label{params-c-s} C_2={c\over s}, \qquad c=\lcm(2,q), \qquad s= \mean\limits_{d\,=\,nc \atop n\in\mathbb N} \, \prod_{p|d\atop p>2}\frac{p-1}{p-2} = \Pi_2^{-1} \, \prod_{p|q\atop p>2}\frac{p}{p-1}, \end{equation} and $\Pi_2={\prod\limits_{p>2}{p(p-2)\over(p-1)^2}}\approx0.66016$ is the twin prime constant. In particular, $s=\Pi_2^{-1}\approx1.51478$ if $q$ is a power of 2; see {\it Appendix}. Note that \begin{equation}\label{tauzero} \tau_{q,r}(d,x) = 0 \quad\mbox{ if } 2\nmid d \mbox{ ~or~ } q\nmid d. \end{equation} Let ${\cal T}_{d;q,r}$ denote the set of primes $p,p'$ at both ends of the gaps $d$ counted in (\ref{main2}): \begin{equation} {\cal T}_{d;q,r} \ =\ \{p, p': ~ p' - p=d ~\mbox{ and }~ p, p' \equiv r \!\!\!\! \mod q\}. \end{equation} Consider the sum of reciprocals of all primes in ${\cal T}_{d;q,r}$: \begin{equation} B_d(q,r)\,=\sum_{p\,\in {\cal T}_{d;q,r}} \frac{1}{p}. \label{def_Bd} \end{equation} We adopt the rule that if a given gap $d$ appears twice in a row: $d=p'-p=p''-p'$, then the corresponding middle prime $p'$ is counted twice. The real numbers $B_d(q,r)$ defined by (\ref{def_Bd}) can be called {\it the generalized Brun constants} for residue class $r$ (mod $q$). Let us define the partial (finite) sums: \begin{equation}\label{Bdxqr} B_d(x;q,r)\,=\sum_{p\in \TT_{d;q,r}\atop p\le x} \frac{1}{p}. \end{equation} Hereafter we will work with rough estimates of partial sums (\ref{Bdxqr}). To distinguish the actual partial sum $B_d(x;q,r)$ from its estimated value, we will write $\BB_d(x;q)$ for the estimate. Likewise, for a rough estimate of the constant (\ref{def_Bd}), we will write $\BB_d(\infty;q)$. Our estimates ${\cal B}_d(\,\cdot\,;q)$ will be independent of $r$. Given a gap size $d=p'-p$, observe that the density of pairs $(p,p')\subset\TT_{d;q,r}$ near $x$ is roughly $\tau_{q,r}(d,x)/x$. Putting $\pi(x)\approx x/\log(x)$ in equation (\ref{main2}), we obtain \begin{equation}\label{integralB} \BB_d(x;q) \approx \sum_{p\in \TT_{d;q,r}} \frac{1}{p} ~- \sum_{p\in \TT_{d;q,r}\atop p>x} \frac{1}{p} \approx \BB_d(\infty,q) - 2C_2 \prod_{p\mid d} \frac{p-1}{p-2}\int_x^\infty \!\!\frac{e^{-{d\over\varphi(q)\log u}}}{\varphi^2(q)\,u\log^2 u}\, {\rm d}u. \end{equation} After the substitution $v={d \over \varphi(q)\log u}$ the integral (\ref{integralB}) can be calculated explicitly: \begin{equation} \BB_d(x;q) \approx \BB_d(\infty;q) - \frac{2C_2}{\varphi(q)d} \prod_{p\mid d} \frac{p - 1}{p - 2} \left(1-e^{-{d\over\varphi(q)\log x}}\right). \label{Bd_x} \end{equation} We will require that $\BB_d(x;q)\to0$ as $x\to1^+$ (indeed, the actual partial sum $B_d(x;q,r)$ will be zero up to the first occurrence of the gap $d$). Taking the limit $x\to 1^+$ in (\ref{Bd_x}) we obtain \begin{equation} \BB_d(\infty;q) \approx \frac{2C_2}{\varphi(q) d} \prod_{p\mid d} \frac{p - 1}{p - 2}. \label{main-B-d} \end{equation} Then the estimate $\BB_d(x;q)$, as a function of $x$ for fixed $q$ and $d$, has the form \begin{equation} \BB_d(x;q) \approx \frac{2C_2}{\varphi(q)d} \prod_{p\mid d} \frac{p - 1}{p - 2} \,e^{-{d\over\varphi(q)\log x}}. \label{Bd-od-x} \end{equation} Recall that, for $d=n\cdot\lcm(2,q)$, the mean value of $\prod_{p\mid d} \frac{p - 1}{p - 2}$ is $s$; see (\ref{main2}), (\ref{params-c-s}). Therefore, we can skip the ratio $\prod_{p\mid d} \frac{p - 1}{p - 2}/s$ (which is near 1 infinitely often) and get \begin{equation} \BB_d(x;q) \approx \frac{2c}{\varphi(q)d} \, e^{-{d\over\varphi(q)\log x}} = \frac{A}{d} \, e^{-{d\over\varphi(q)\log x}}, \end{equation} where $c=\lcm(2,q)$ and $A=2c/\varphi(q)=O(\log\log q)$; moreover\footnote{ We easily check that $A=4$ when $q=2$, and $A\le6$ when $q$ is an odd prime. On the other hand, one can also show that $A=O(\log\log q)$ when $q$ is a primorial (which constitutes an extreme case; see \cite{Nicolas1983}). } $A=O(1)$ for prime $q$. Suppose that gap $d$ occurs for the first time at $\xi=P_f(d;q,r)$. We can estimate $\xi$ from the condition $\BB_d(\xi;q)\approx2/\xi$, so we have the following equation for $\xi$: \begin{equation}\label{Aexpd2xi} \frac{A}{d} \, e^{-{d\over\varphi(q)\log\xi}} = \frac{2}{\xi}. \end{equation} Taking the log of both sides of (\ref{Aexpd2xi}) and discarding the small term $\log{A\over2}$, we get a quadratic equation for $t=\log\xi$: \begin{equation}\label{quadratic} t^2 - t\log d - {d/\varphi(q)} = 0. \end{equation} Its positive solution is $t\approx{1\over2}\log d+\sqrt{d/\varphi(q)}$, which allows us to conjecture that \begin{equation}\label{dexpsqrtdphi} \xi = P_f(d;q,r) \asymp \sqrt{d} \, e^{\sqrt{d/\varphi(q)}} \end{equation} infinitely often. \begin{remark} Setting $q=2$ in (\ref{dexpsqrtdphi}), for the first occurrence of gap $d$ in the sequence of all primes we get $p_f(d) \asymp \sqrt{d} \, e^{\sqrt{d}}$; cf.\,\cite{Wolf1997}. Thus, in progressions (P) with large $q$, gaps of a given size $d$ appear much earlier (due to the division by $ \varphi(q)$ under square root in the exponent). Note, however, that the ``natural unit'' of gaps in progressions (P) is larger than that for all primes; namely, all even gaps between primes in (P) have sizes $n\cdot\lcm(2,q)$, $n\in{\mathbb N}$. \end{remark} \section{How many first-occurrence gaps are there below $x$?} Using a modified version of our {\tt PARI/GP} code from \cite{KourbatovWolf2019} we computed the mean numbers of first-occurrence gaps between primes $p=r+nq$ in intervals $[x,ex]$, $x=e^j$, $j=1,2,\ldots,27$. Figure \ref{figq17011meanfog} shows the results of this computational experiment for $q=17011$. For comparison, we also computed the corresponding mean numbers of record (maximal) gaps between primes $p=r+nq$ in the same set of intervals $[x,ex]$; the results are shown in Fig.\,\ref{figq17011meanrec}. The computation resulting in Figs.\,\ref{figq17011meanfog}, \ref{figq17011meanrec} took about 30 hours on an Intel Core i7-8550U CPU. We found that, for large enough $x$, the mean number of first-occurrence gaps between primes $r+nq\in[x,ex]$ grows like a linear function of $\log x$; see Fig.\,\ref{figq17011meanfog}. It is reasonable to expect that the {\it total} number $N'_{q,r}(x)$ of first-occurrence gaps between primes $\le x$ in progression (P): $r+nq$ can be approximated by a quadratic function of $\log x$, and for large $x$ we will have \begin{equation} N'_{q,r}(x) \approx {T_0(q,x)\over\lcm(2,q)} \lesssim {\varphi(q)\log^2x\over\lcm(2,q)} = O(\log^2x), \end{equation} where $T_0(q,x)$ is given by (\ref{T0qx}). \begin{figure}[H] \begin{center} \includegraphics[width=5in,height=4.6in,keepaspectratio]{akmw2020-q17011-meanfogx} \caption{Mean number of first-occurrence gaps between primes $p=r+nq\in[x,ex]$,\, for $q=17011$, $x=e^j$, $j\le27$. Averaging for all $r\in[1,17010]$. } \label{figq17011meanfog} \vspace*{8mm} \includegraphics[width=5in,height=2.94in,keepaspectratio]{akmw2020-q17011-meanrecx} \caption{Mean number of record (maximal) gaps between primes $p=r+nq\in[x,ex]$,\, for $q=17011$, $x=e^j$, $j\le27$. Averaging for all $r\in[1,17010]$. } \label{figq17011meanrec} \end{center} \end{figure} Thus the number $N'_{q,r}(x)$ of first-occurrence gaps between primes in (P) is asymptotically {\it much greater} than the number $N_{q,r}(x)$ of record (maximal) gaps in (P). Indeed, the latter seems to be only $O(\log x)$. More precisely, our computations (see Fig.\,\ref{figq17011meanrec}) suggest that the average number of maximal gaps observed in $[x,ex]$ is \begin{equation}\label{eq-hyperbola} \mean\limits_{r} \big( N_{q,r}(ex) - N_{q,r}(x) \big) \approx\, 2 - {\kappa(q) \over \log x + \delta(q)}. \end{equation} Accordingly, we conjecture that the number of maximal gaps between primes $\le x$ in (P) is \begin{equation}\label{eq-2logx} N_{q,r}(x) \sim 2\log x = O(\log x) \qquad\mbox{ as }x\to\infty. \end{equation} The constant 2 in (\ref{eq-hyperbola}) and (\ref{eq-2logx}) can be justified by a heuristic argument based on two hypotheses: the generalized Shanks conjecture (\ref{genShanks}) and the assumption that, on average, consecutive record gaps near $x$ differ by the {\it average} gap $\sim \varphi(q)\log x$ between primes in (P). (See \cite{Kourbatov2018, KourbatovWolf2019} for details.) \section{Maximal gaps and first occurrences as functions of their sequential number}\label{S7} As before, consider the arithmetic progression (P): \ $r, \ r+q, \ r+2q, \ r+3q,\,\ldots$~. \begin{conjecture}\label{conj-secfopgap} Let $S(n;q,r)$ be the size of the $n$-th first-occurrence gap between primes in progression (P). Then the sizes $S(n;q,r)$ of almost all first-occurrence gaps satisfy \begin{equation}\label{dblSnqr} n \le S(n;q,r) \le 2nq\lceil \log^2 q \rceil. \end{equation} Moreover, all gap sizes divisible by $\lcm(2,q)$ occur early enough, so that \begin{equation} S(n;q,r) \approx n\cdot\lcm(2,q) = O_q(n) \qquad\mbox{ as }n\to\infty. \end{equation} {\it Example}: for $q=6$ and $r=5$, we have $S(n;6,5)=$\,\seqnum{A334543}$(n)$. \end{conjecture} For comparison, here is a similar conjecture for {\it maximal} gap sizes: \begin{conjecture}\label{conj-secmaxgap} Let $R(n;q,r)$ be the size of the $n$-th record (maximal) gap between primes in (P). Then the sizes $R(n;q,r)$ of almost all maximal gaps satisfy \cite{Kourbatov2018,puzzle_84} \begin{equation}\label{dblRnqr} {\varphi(q) n^2 \over 6} < R(n;q,r) < \varphi(q) n^2 + (n+2)q\log^2 q. \end{equation} \end{conjecture} We do not know any exceptions to (\ref{dblRnqr}); in particular, there are no exceptions for $q\le2000$, $n<15$. Heuristic reasoning \cite{Kourbatov2018, KourbatovWolf2019} suggests that for large $n$ we will have \begin{equation} R(n;q,r) \approx {\varphi(q) n^2 \over 4} = O_q(n^2) \qquad\mbox{ as }n\to\infty. \end{equation} (At the same time, for small $n$, the ratio $R(n;q,r)\over\varphi(q) n^2$ may be greater than 1.) \itpara{Example}: for $q=6$ and $r=5$, we have $R(n;6,5)=$\,\seqnum{A268928}$(n)$. Conjectures \ref{conj-secfopgap} and \ref{conj-secmaxgap} are another way to restate our observation that first-occurrence gaps between primes in progression (P) are much more numerous than maximal gaps. \begin{remark} Setting $n=0$ in (\ref{dblRnqr}), we get that the ``0th record gap'' is less than $2q \log^2 q$, which can be interpreted as an almost-sure upper bound on the least prime in progression (P); cf.\,\cite[sect.2]{LiPrattShakan2017}. That is, the ``0th record gap'' between primes in (P) is the distance from 0 to the least prime in (P), which is (almost always) bounded by $2q \log^2 q$. \end{remark} \section{Appendix} In this appendix we calculate the average value of the product \begin{equation}\label{defSd} S(d) = \prod_{p|d\atop p>2} {p-1\over p-2} \end{equation} when $d$ runs through a given arithmetic progression, $d=r+nq$, $n\in\mathbb N$. So our goal is to find $$ \mean\limits_{d=r+nq\atop n\in\mathbb N} S(d) = \lim_{N\to\infty} {1\over N} \sum_{n=1}^N S(r+nq). $$ For generality, in the appendix, $q$ and $r$ are not necessarily coprime, and $r$ is nonnegative. In particular, we will derive the expression for $s$ in (\ref{params-c-s}) for the case $d=nq$ \ ($r=0$). Bombieri and Davenport \cite{Bombieri} proved that $$ \mean\limits_{d\in\mathbb N} S(d) = \mean\limits_{d\in\mathbb N}\prod\limits_{p|d\atop p>2} {p-1\over p-2} = \Pi_2^{-1} = 1.51478\ldots\,, $$ where $\Pi_2$ is the {\it twin prime constant} \seqnum{A005597}: $$ \Pi_2=\prod\limits_{p>2}{p(p-2)\over(p-1)^2} = 0.66016\ldots\,. $$ If, instead of averaging $S(d)$ over all natural numbers, we restrict the averaging process to an arithmetic progression $r+nq$, then the resulting average might become larger or smaller than $\Pi_2^{-1}$---or it might remain unchanged ($=\Pi_2^{-1}$). A few special cases are possible, depending on the values of $q$ and $r$. \subsection{Base cases: prime $q$}\label{appendix-primeq} (i) If $q=2$, we have $S(2n)=S(n)$ for all $n \in\mathbb N$. Therefore, for even $d$, $$\mean\limits_{d=2n}S(d)=\mean\limits_{d\in\mathbb N}S(d)=\Pi_2^{-1}. $$ But this immediately implies that, for odd $d$, we must also have $$\mean\limits_{d=2n+1}S(d)=\mean\limits_{d\in\mathbb N}S(d)=\Pi_2^{-1}. $$ \rmpara{(ii)} Suppose $q=p'$ is an odd prime and $\gcd(p',r)=1$. Let \begin{equation*} \begin{split} \xi &\,= \xi(p') = \,\mean\limits_{d=np'} S(d) \\ \eta &\,= \eta(p') = \mean\limits_{d=r+np'} S(d). \end{split} \end{equation*} Observe that every term in progression $d=np'$ has the prime factor $p'$; so the factor ${p'-1\over p'-2}$ is {\it always present} in the corresponding product (\ref{defSd}). On the other hand, terms in progression $d=r+np'$ never have the prime factor $p'$; so the factor ${p'-1\over p'-2}$ is {\it never present} in the corresponding product (\ref{defSd}). All other odd prime factors $p$ (resp., ${p-1\over p-2}$), $p\ne p'$, are {\it sometimes} present in both progressions (resp., products), with probability $1/p$. Therefore, we require that \begin{equation}\label{xi-eta-S} \xi=\eta\cdot{p'-1\over p'-2}. \end{equation} The factor ${p'-1\over p'-2}$ in (\ref{xi-eta-S}) is the same as in the definition of $S(d)$. Note that there are $p'-1$ residue classes mod $p'$ for which the mean values of $S(d)$ are $\eta$, and only one ``special'' residue class (0 mod $p'$) for which the mean value of $S(d)$ is $\xi$. But the arithmetic average of all these $p'$ mean values is equal to $\mean\limits_{d\in\mathbb N} S(d)$, which is $\Pi_2^{-1}$: \begin{equation}\label{weighted-avg} {1\over p'}\cdot\xi + {p'-1\over p'}\cdot \eta = \Pi_2^{-1}. \end{equation} Solving equations (\ref{xi-eta-S}) and (\ref{weighted-avg}) together, we find \begin{align}\label{xi-formula} \xi &= \mean\limits_{d=np'} \,S(d) = {p'\over p'-1} \cdot \Pi_2^{-1}, \\ \eta &=\!\mean\limits_{d=r+np'} \!S(d) = {p'(p'-2)\over(p'-1)^2} \cdot \Pi_2^{-1}. \label{eta-formula} \end{align} {\it Example}: for $q=3$, we have $$\mean\limits_{d=3n} S(d)= \displaystyle{3\over2}\Pi_2^{-1}, \qquad \mean\limits_{d=3n+1} S(d)= \mean\limits_{d=3n+2} S(d)= \displaystyle{3\over4}\Pi_2^{-1}. $$ Formulas (\ref{xi-formula}) and (\ref{eta-formula}) encode the following \bfpara{Factors Principle.} For progressions $d=np'$ where each term is guaranteed to have a factor $p'$, the average product $S(d)$ equals $\Pi_2^{-1}$ multiplied by ${p'\over p'-1}>1$. For progressions $d=r+np'$ where each term {\it never} has the factor $p'$, the average is $\Pi_2^{-1}$ multiplied by ${p'(p'-2)\over(p'-1)^2}<1$. \subsection{Remaining cases: composite $q$} The above argument considered progressions with one ``special'' prime factor $p'$. A straightforward modification of the argument allows us to extend the consideration from progressions with $k$ ``special'' factors to those with $k+1$ ``special'' factors. Therefore, we can apply the ``factors principle'' to composite values of $q$. \rmpara{(A)} Let $q$ be a power of 2. (This case is similar to (i) in sect.\,\ref{appendix-primeq}.) In this case, for every odd prime $p$, terms of our arithmetic progression $r+nq$ span {\it all residue classes} modulo $p$. So here the average value of $S(d)$ remains unchanged: $$\mean\limits_{d=r+qn} S(d)=\Pi_2^{-1}. $$ {\it Example}: for $q=4$, we have $\mean\limits_{d=4n} S(d)=\mean\limits_{d=4n+1} S(d)= \mean\limits_{d=4n+2} S(d)=\mean\limits_{d=4n+3} S(d)= \Pi_2^{-1}$. \rmpara{(B)} Suppose $q\ne2^m$ is composite and $\gcd(q,r)=1$. Here all terms of our arithmetic progression $r+nq$ are not divisible by prime factors of $q$, while divisibility by all other primes is neither ensured nor precluded. We have $$\mean\limits_{d=r+nq} S(d)=\Pi_2^{-1}\cdot \prod\limits_{p|q\atop p>2}{p(p-2)\over(p-1)^2}. $$ {\it Example}: for $q=15$ and $r=1$, we have $\mean\limits_{d=15n+1} S(d)=\displaystyle{3\over4}\cdot{15\over16}\cdot \Pi_2^{-1}=\displaystyle{45\over64} \Pi_2^{-1}$. \rmpara{(C)} Suppose $q\ne2^m$ is composite and $r=0$. All terms $nq$ in our arithmetic progression are divisible by prime factors of $q$, while divisibility by all other primes is neither ensured nor precluded. We have $$\mean\limits_{d=nq} S(d)=\Pi_2^{-1}\cdot \prod\limits_{p|q\atop p>2}{p\over p-1}. $$ {\it Example}: for $q=15$ and $r=0$, we have $\mean\limits_{d=15n} S(d)=\displaystyle{3\over2}\cdot{5\over4}\cdot \Pi_2^{-1}=\displaystyle{15\over8} \Pi_2^{-1}$. \rmpara{(D)} Suppose $q\ne2^m$ is composite, $r\ne0$, $\gcd(q,r)>1$. (This case is, in a sense, a combination of the above cases.) Terms of our arithmetic progression $r+nq$ are divisible by common prime factors of $q$ and $r$, but not by other prime factors of $q$; divisibility by all other primes is neither ensured nor precluded. We have \begin{equation}\label{meanSd-general} \mean\limits_{d=r+nq} S(d)=\Pi_2^{-1} \cdot \prod\limits_{P|q,\,P|r\atop P>2}\!{P\over P-1}\, \cdot \prod\limits_{p|q,\,p\,\nmid\, r\atop p>2} \!{p(p-2)\over (p-1)^2}. \end{equation} {\it Example}: for $q=30$ and $r=3$, we have $\mean\limits_{d=30n+3} S(d)=\displaystyle{3\over2}\cdot{15\over16}\cdot \Pi_2^{-1}=\displaystyle{45\over32} \Pi_2^{-1}$. \begin{remark} Formula (\ref{meanSd-general}) is in fact applicable to an arbitrary pair $(q,r)$; it subsumes all preceding cases. Formulas like this can be derived not only in our setup related to the twin prime constant $\Pi_2$ but also for similar average products related to the Hardy--Littlewood $k$-tuple constants, as well as for other average products, e.g., $\mean\limits_{d=r+nq \atop n\in{\mathbb N}} \prod\limits_{p|d\atop p>m}{p+a\over p+b}$. \end{remark} \section{Acknowledgments} We are grateful to the anonymous referee and to all contributors and editors of the websites {\tt MersenneForum.org}, {\tt OEIS.org}, and {\tt PrimePuzzles.net}. Special thanks to Martin Raab, whose extensive computations provided new examples of exceptionally large gaps between primes in residue classes \cite{Raab2020}--\cite{Raab2020may}. We gratefully acknowledge the valuable input and encouragement of the late Prof.~Thomas R.~Nicely, whose emails we retain to this day. \begin{thebibliography}{99} \bibitem{BakerHarmanPintz} R. C. Baker, G. Harman, and J. 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Comp.} {\bf 52} (1989), 221--224. \end{thebibliography} \bigskip \hrule \bigskip \noindent {\it 2010 Mathematics Subject Classification}: Primary 11N05; secondary 11N56, 11N64. \par \noindent {\it Keywords}: arithmetic progression, Brun constant, Cram\'er conjecture, Gumbel distribution, prime gap, residue class, Shanks conjecture. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A000010}, \seqnum{A001620}, \seqnum{A005597}, \seqnum{A014320}, \seqnum{A065421}, \seqnum{A167864}, \seqnum{A194098}, \seqnum{A268925}, \seqnum{A268928}, \seqnum{A330853}, \seqnum{A330854}, \seqnum{A330855}, \seqnum{A334543}, \seqnum{A334544}, \seqnum{A334545}, \seqnum{A335366}, and \seqnum{A335367}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received June 11 2020; revised version received September 15 2020, September 19 2020. Published in {\it Journal of Integer Sequences}, October 15 2020. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{https://cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}