\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \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,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://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} \begin{center} \vskip 1cm{\LARGE\bf Formulas for Odd Zeta Values \vskip .1in and Powers of $\pi$ } \vskip 1cm \large Marc Chamberland and Patrick Lopatto\\ Department of Mathematics and Statistics\\ Grinnell College\\ Grinnell, IA 50112\\ USA \\ \href{mailto:chamberl@math.grinnell.edu}{\tt chamberl@math.grinnell.edu} \\ \end{center} \vskip .2 in \begin{abstract} Plouffe conjectured fast converging series formulas for $\pi^{2n+1}$ and $\zeta (2n+1)$ for small values of $n$. We find the general pattern for all integer values of $n$ and offer a proof. \end{abstract} \makeatletter \newcommand{\rmnum}[1]{\romannumeral #1} \newcommand{\Rmnum}[1]{\expandafter\@slowromancap\romannumeral #1@} \newcommand{\qdp}{\hspace*{1.5mm}}% \newcommand{\qqdp}{\hspace*{2.5mm}} \newcommand{\xqdp}{\hspace*{5.0mm}} \newcommand{\xxqdp}{\hspace*{10mm}} \newcommand{\xquad}{\quad\qquad\quad} \newcommand{\qdn}{\hspace*{-1.5mm}} \newcommand{\qqdn}{\hspace*{-2.5mm}} \newcommand{\xqdn}{\hspace*{-5.0mm}} \newcommand{\xxqdn}{\hspace*{-10mm}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\fns}{\footnotesize} \newcommand{\dst}{\displaystyle} \newcommand{\sst}{\scriptstyle} \newcommand{\sss}{\scriptscriptstyle} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\+}{&\qqdn}% \newcommand{\zero}{&\qdn&\xqdn} \newcommand{\tagg}[2]{\addtocounter{equation}{#1} \tag{\theequation#2}}%%%\tag{}\tag*{} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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%%%%%%%%%%%%%%%%%%%%% positions %%%%%%%%%%%%%%%%%%%%%% \newcommand{\centro}[1] {\begin{center}#1\end{center}} \newcommand{\sinistra}[1] {\begin{flushleft}#1\end{flushleft}} \newcommand{\destra}[1] {\begin{flushright}#1\end{flushright}} \newcommand{\centerbox}[2]{\centro{\begin{tabular}{|c|}\hline \parbox{#1}{\mbox{}\\[2.5mm]#2\mbox{}\\}\\\hline\end{tabular}}} \newcommand{\centrobmp}[3]{\vspace*{1mm} \centerbmp{#1}{#2}{#3}\vspace*{1mm}} \newcommand{\sav}[1]{\newsavebox{#1}% \sbox{#1}{\theequation}\\[-2mm]}%=\newline \newcommand{\usa}[1]{(\usebox{#1})}%{#1}={\some-nome} \newcommand{\summary}[2]{\begin{center}\parbox{#1} {{\sc\bf Summary}:\,{\small\it#2}}\end{center}} \newcommand{\riassunto}[2]{\begin{center}\parbox{#1} {{\sc\bf Riassunto}:\,{\small\it#2}}\end{center}} \newcommand{\grazia}[2]{\begin{center}\parbox{#1} {{\sc\bf Ringraziamento}:\,{\small\it#2}}\end{center}} \newcommand{\thank}[2]{\begin{center}\parbox{#1} {{\sc\bf Acknowledgement}:\,{\small\it#2}}\end{center}} \newcommand{\bbtm}[5]{\bibitem{kn:#1}{#2,}~{#3,}~\emph{#4}~{#5.}} \newcommand{\bbtmn}[4]{\bibitem{kn:#1}{#2,}~\emph{#3,}~{#4.}} \newcommand{\bbtmm}[4]{\bibitem{kn:#1}{#2,}~{#4.}} \newcommand{\cito}[1]{\cite{kn:#1}} \newcommand{\citu}[2]{\cite[#2]{kn:#1}} \newcommand{\mr}{\textbf{MR}\:} \newcommand{\zbl}{\textbf{Zbl}\:} \newcommand{\nota}[2]{\centerbox{#1}{\textbf{Achtung}\quad#2}} \newcommand{\graph}[3]{\begin{picture}(0,0) \put(#1,#2){#3}\end{picture}} \newcommand{\fwd}{\mbox{$\bigtriangleup\qdn\!\cdot\,\:$}} \newcommand{\bwd}{\bigtriangledown} \section{Introduction} \label{sec-intro} \setcounter{equation}{0} It took nearly one hundred years for the Basel Problem --- finding a closed form solution to $\sum_{k=1}^\infty 1/k^2$ --- to see a solution. Euler solved this in 1735 and essentially solved the problem where the power of two is replaced with any even power. This formula is now usually written as $$ \zeta (2n ) = (-1)^{n+1} \frac{ B_{2n}(2\pi)^{2n} }{ 2(2n)! } , $$ where $\zeta(s)$ is the Riemann zeta function and $B_k$ is the $k^{th}$ Bernoulli number uniquely defined by the generating function $$ \frac{x}{e^x - 1} = \sum_{n=0}^\infty \frac{B_n x^n}{n!}, ~~~|x|<2\pi. $$ and whose first few values are $0,-1/2, 1/6, 0, -1/30, \ldots$. However, finding a closed form for $\zeta(2n+1)$ has remained an open problem. Only in 1979 did Ap\'ery show that $\zeta (3)$ is irrational. His proof involved the snappy acceleration $$ \zeta (3) = \frac{5}{2}\sum_{n=1}^\infty \frac{ (-1)^{n-1} }{ n^3 {2n\choose n}} . $$ This tidy formula does not generalize to the other odd zeta values, but other representations, such as nested sums or integrals, have been well-studied. The hunt for a clean result like Euler's has largely been abandoned, leaving researchers with the goal of finding formulas which either converge quickly or have an elegant form. Following his success in discovering a new formula for $\pi$, Simon Plouffe\cite{Plouffe} postulated several identities which relate either $\pi^m$ or $\zeta (m)$ to three infinite series. Letting $$ S_n (r) = \sum_{k=1}^\infty \frac{1}{k^n (e^{\pi rk} - 1)}, $$ the first few examples are \begin{eqnarray*} \pi & = & 72 S_1 (1) - 96 S_1 (2) + 24 S_1 (4) \\ \pi^3 & = & 720 S_3 (1) - 900 S_3 (2) + 180 S_3 (4) \\ \pi^5 & = & 7056 S_5 (1) - 6993 S_5 (2) + 63 S_5 (4) \\ \pi^7 & = & \frac{907200}{13} S_7 (1)-70875 S_7 (2)+\frac{14175}{13} S_5 (4) \end{eqnarray*} and \begin{eqnarray*} \zeta (3) & = & 28 S_3 (1) - 37 S_3 (2) + 7 S_3 (4) \\ \zeta (5) & = & 24 S_5 (1) - \frac{259}{10} S_5 (2) + \frac{1}{10} S_5 (4) \\ \zeta (7) & = & \frac{304}{13} S_7 (1) - \frac{103}{4} S_7 (2) + \frac{19}{52} S_7 (4) . \end{eqnarray*} Plouffe conjectured these formulas by first assuming, for example, that there exist constants $a$, $b$, and $c$ such that $$ \pi = a S_1 (1) + b S_1 (2) + c S_1 (4). $$ By obtaining accurate approximations of each the three series, he wrote some computer code to postulate rational values for $a,b,c$. Today, such integer relations algorithms have been used to discover many formulas. The widely used PSLQ algorithm, developed by Ferguson and Bailey\cite{FB}, is implemented in Maple. The following Maple code (using Maple 14) solves the above problem: \begin{verbatim} > with(IntegerRelations): > Digits := 100; > S := r -> sum( 1/k/( exp(Pi*r*k)-1 ), k=1..infinity ); > PSLQ( [ Pi, S(1), S(2), S(4) ] ); \end{verbatim} The PSLQ command returns the vector $[-1,72,-96, 24]$, producing the first formula. While the computer can be used to conjecture the coefficients of a specific power, finding the general sequence of rationals has remained an open problem. This note finds the sequences and offers formal proofs. \section{ Exact Formulas } While it does not seem that $\zeta(2n+1)$ is a rational multiple of $\pi^{2n+1}$, a result in Ramanujan's notebooks gives a relationship with infinite series which converge quickly. % We define % \begin{equation*} % S_{n}(r)=\sum_{k=1}^{\infty}\frac{1}{k^{n}(e^{r k}-1)} % \end{equation*} % and let $B_j$ denote the $j^{th}$ Bernoulli number. \begin{theorem} (Ramanujan) \label{Rama_thm} If $\alpha>0$, $\beta>0$, and $\alpha \beta = \pi^2$, then % \begin{multline} % \alpha^{-n}\left\{\frac{1}{2}\zeta(2n+1)+S_{2n+1}(2\alpha) \right\} = \\(-\beta)^{-n}\left\{\frac{1}{2}\zeta(2n+1)+S_{2n+1}(2\beta) \right\} - 4^n\sum_{k=0}^{n+1}(-1)^k\frac{B_{2k} B_{2n+2-2k}}{(2k)!(2n+2-2k)!}\alpha^{n+1-k}\beta^k % \end{multline} \begin{eqnarray*} \lefteqn{ \alpha^{-n}\left\{\frac{1}{2}\zeta(2n+1)+S_{2n+1}(2\alpha) \right\} = }\\ & & (-\beta)^{-n}\left\{\frac{1}{2}\zeta(2n+1)+S_{2n+1}(2\beta) \right\} - 4^n\sum_{k=0}^{n+1}(-1)^k\frac{B_{2k} B_{2n+2-2k}}{(2k)!(2n+2-2k)!}\alpha^{n+1-k}\beta^k . \end{eqnarray*} \end{theorem} Using $\alpha=\beta=\pi$ in Proposition \ref{Rama_thm} and defining $$ F_{n}=\sum_{k=0}^{n+1}(-1)^k\frac{B_{2k} B_{2n+2-2k}}{(2k)!(2n+2-2k)!} , $$ we have $$ \left( \pi^{-n} - (-\pi)^{-n} \right) \left(\frac{1}{2}\zeta(2n+1)+S_{2n+1}(2\pi) \right) = - 4^n\pi^{n+1}F_n . $$ To find formulas for the odd zeta values and powers of $\pi$, we will divide these into two classes: $\zeta (4m-1)$ and $\zeta (4m+1)$. Such distinctions can be seen in other studies; see \cite[pp. 137--139]{BB}. First we find the formulas for $\pi^{4m-1}$ and $\zeta(4m-1)$. If $n$ is odd, then \begin{equation} \label{eqn_(3)} \frac{1}{2}\zeta(2n+1)+S_{2n+1}(2\pi) = \frac{- 4^n}{2}\pi^{2n+1}F_n \end{equation} Using $\alpha=\pi/2$ and $\beta=2\pi$ in Proposition \ref{Rama_thm} and defining $$ G_n =\sum_{k=0}^{n+1}(-4)^k\frac{B_{2k} B_{2n+2-2k}}{(2k)!(2n+2-2k)!} , $$ % \begin{multline*} % \frac{\pi^{-n}}{2^{-n}}\left\{\frac{1}{2}\zeta(2n+1)+S_{2n+1}(\pi) \right\} = \\ {(-2\pi)}^{-n}\left\{\frac{1}{2}\zeta(2n+1)+S_{2n+1}(4\pi) \right\} - 2^{-n-1}4^n\pi^{n+1}\sum_{k=0}^{n+1}(-4)^k\frac{B_{2k} B_{2n+2-2k}}{(2k)!(2n+2-2k)!} % \end{multline*} % % \begin{multline*} % -2^n\left\{\frac{1}{2}\zeta(2n+1)+S_{2n+1}(\pi) \right\} = \\ 2^{-n}\left\{\frac{1}{2}\zeta(2n+1)+S_{2n+1}(4\pi) \right\} +2^{-n-1}4^n\pi^{2n+1}\sum_{k=0}^{n+1}(-4)^k\frac{B_{2k} B_{2n+2-2k}}{(2k)!(2n+2-2k)!} % \end{multline*} % % % Now we simplify: % \begin{multline*} % -2^n\left\{\frac{1}{2}\zeta(2n+1)+S_{2n+1}(\pi) \right\} = \\ 2^{-n}\left\{\frac{1}{2}\zeta(2n+1)+S_{2n+1}(4\pi) \right\} +2^{-n-1}4^n\pi^{2n+1}G_n % \end{multline*} % % \begin{equation*} % -4^n\left\{\frac{1}{2}\zeta(2n+1)+S_{2n+1}(\pi) \right\} = \left\{\frac{1}{2}\zeta(2n+1)+S_{2n+1}(4\pi) \right\} + \frac{4^n}{2}\pi^{2n+1}G_n % \end{equation*} % % \begin{equation*} % \frac{-4^n}{2}\zeta(2n+1)-4^nS_{2n+1}(\pi) = \frac{1}{2}\zeta(2n+1)+S_{2n+1}(4\pi) + \frac{4^n}{2}\pi^{2n+1}G_n % \end{equation*} % % \begin{equation*} % \frac{-4^n}{2}\zeta(2n+1) = \frac{1}{2}\zeta(2n+1)+S_{2n+1}(4\pi) +4^nS_{2n+1}(\pi) + \frac{4^n}{2}\pi^{2n+1}G_n % \end{equation*} % % % \begin{equation*} % \frac{-(4^n+1)}{2}\zeta(2n+1) = S_{2n+1}(4\pi) +4^nS_{2n+1}(\pi) + \frac{4^n}{2}\pi^{2n+1}G_n % \end{equation*} % one has $$ \frac{1}{2}\zeta(2n+1) = \frac{S_{2n+1}(4\pi) +4^nS_{2n+1}(\pi) + \frac{4^n}{2}\pi^{2n+1}G_n}{-(4^n+1)} . $$ Combining this with equation (\ref{eqn_(3)}) yields % \begin{equation*} % \frac{S_{2n+1}(4\pi) +4^nS_{2n+1}(\pi) + \frac{4^n}{2}\pi^{2n+1}G_n}{-(4^n+1)}+S_{2n+1}(2\pi) = \frac{- 4^n}{2}\pi^{2n+1}F_n % \end{equation*} % % \begin{equation*} % S_{2n+1}(4\pi) +4^nS_{2n+1}(\pi) + \frac{4^n}{2}\pi^{2n+1}G_n-(4^n+1)S_{2n+1}(2\pi) = \frac{4^n}{2}\pi^{2n+1}(4^n+1)F_n % \end{equation*} % % \begin{equation*} % 4^nS_{2n+1}(\pi) -(4^n+1)S_{2n+1}(2\pi) +S_{2n+1}(4\pi)= \frac{4^n}{2}\pi^{2n+1}(4^n+1)F_n-\frac{4^n}{2}\pi^{2n+1}G_n % \end{equation*} % % \begin{equation*} % 4^nS_{2n+1}(\pi) -(4^n+1)S_{2n+1}(2\pi) +S_{2n+1}(4\pi)= \pi^{2n+1}\left(\frac{ 4^n}{2}(4^n+1)F_n-\frac{4^n}{2}G_n\right) % \end{equation*} % $$ \frac{4^nS_{2n+1}(\pi) -(4^n+1)S_{2n+1}(2\pi) +S_{2n+1}(4\pi)}{\frac{4^n}{2}(4^n+1)F_n-\frac{4^n}{2}G_n}= \pi^{2n+1} $$ Substituting $n=2m-1$ and defining $$ D_m = 4^{2m-1} [ (4^{2m-1}+1)F_{2m-1} - G_{2m-1} ]/2 $$ produces % \begin{equation*} % \frac{4^{2m-1}S_{4m-1}(\pi) -(4^{2m-1}+1)S_{4m-1}(2\pi) +S_{4m-1}(4\pi)}{\frac{4^{2m-1}}{2}\{(4^{2m-1}+1)F_n-G_n\}}= \pi^{4m-1} % \end{equation*} % $$ \pi^{4m-1} = \frac{4^{2m-1}}{D_m} S_{4m-1}(\pi) - \frac{4^{2m-1}+1}{D_m} S_{4m-1}(2\pi) + \frac{1}{D_m} S_{4m-1}(4\pi) $$ This identity may be used in conjunction with equation (\ref{eqn_(3)}) to obtain $$ \zeta(4m-1) = -\frac{F_{2m-1} 4^{4m-2} }{D_m} S_{4m-1}(\pi) +\frac{G_{2m-1} 4^{2m-1} }{D_m} S_{4m-1}(2\pi) -\frac{F_{2m-1} 4^{2m-1} }{D_m} S_{4m-1}(4\pi) . $$ % \begin{multline*} % -\frac{2F_n4^nS_{2n+1}(\pi)}{(4^n+1)F_n-G_n}+\frac{ 2{G_n}S_{2n+1}(2\pi) }{(4^n+1)F_n-G_n} % -\frac{2{F_n}S_{2n+1}(4\pi)}{{(4^n+1)F_n-G_n}} = % \\ \zeta(2n+1) % \end{multline*} % % And the desired expression follows from substituting $n=2m-1$, so that $2n+1$ becomes $4m-1$. % To obtain formulas for the $4m+1$ cases, use $\alpha=\pi/2$ and $\beta=2\pi$ in Theorem \ref{Rama_thm} with $n=2m$ to obtain \begin{equation} \label{eqn_(4)} \zeta(4m+1) = \frac{\frac{4^{2m}}{2}\pi^{4m+1}G_{2m} -S_{4m+1}(4\pi)+4^{2m}S_{4m+1}(\pi)}{\frac{1}{2}(1-4^{2m})} . \end{equation} Define $T_n (r)$ (similar to $S_n (r)$) by $$ T_{n}(r)=\sum_{k=1}^{\infty}\frac{1}{k^{n}(e^{r k}+1)} $$ and another finite sum of Bernoulli numbers by $$ H_{n}=\sum_{k=0}^{n}(-4)^{n+k}\frac{B_{4k} B_{4n+2-4k}}{(4k)!(4n+2-4k)!} . $$ Vepstas \cite{Vepstas} cites an expression credited to Ramanujan: \begin{eqnarray*} \label{eqn_Vepstas} \lefteqn{ (1+(-4)^m-2^{4m+1}) \zeta(4m+1)= } \\ & & 2T_{4m+1}(2\pi)+2(2^{4m+1}-(-4)^m)S_{4m+1}(2\pi) +2^{4m+1}\pi^{4m+1}H_m+2^{4m}\pi^{4m+1}G_{2m} . \end{eqnarray*} Vepstas also produces a formula to show the relationship between $T_k$ and $S_k$: $$ T_k(x)=S_k(s)-2S_k(2x) $$ Combining the last two equations produces \begin{eqnarray*} \lefteqn{ \frac{1+(-4)^m-2^{4m+1}} {\frac{1}{2}(1-4^{2m})} \left(\frac{4^{2m}}{2}\pi^{4m+1}G_{2m}-S_{4m+1}(4\pi)+4^{2m}S_{4m+1}(\pi)\right) = }\\ & & 2[2^{4m+1}-(-4)^m+1]S_{4m+1}(2\pi) -4S_{4m+1}(4\pi)+2^{4m+1}\pi^{4m+1}H_m+2^{4m}\pi^{4m+1}G_{2m} \end{eqnarray*} Letting $$ K_m=\frac{\frac{1}{2}(1-4^{2m})}{1+(-4)^m-2^{4m+1}} $$ and $$ E_m = \frac{4^{2m}}{2}G_{2m}-2^{4m+1}K_mH_m-2^{4m}K_mG_{2m} , $$ one eventually finds $$ \pi^{4m+1} = -\frac{4^{2m}}{E_m} S_{4m+1}(\pi) +\frac{ 2K_m[2^{4m+1}-(-4)^m+1]}{E_m} S_{4m+1}(2\pi) +\frac{ (1-4K_m)}{E_m}S_{4m+1}(4\pi) $$ Substituting this into equation (\ref{eqn_(4)}) produces \begin{eqnarray*} \zeta(4m+1) & = & -\frac{ 16^m( G(2m)16^m+2E_m) }{(-1+16^m)E_m} S_{4m+1}(\pi) \\ & & -\frac{ 2G_{2m}K_m 16^m (2\cdot 16^m - (-4)^m + 1) } {(-1+16^m)E_m} S_{4m+1}(2\pi) \\ & & -\frac{ G(2m)16^m + 4 G_{2m} 16^m K_m + 2E_m } {(-1+16^m)E_m} S_{4m+1}(4\pi) . \end{eqnarray*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{9} \bibitem{BB} D. Bailey and J. Borwein. \newblock {\em Experimentation in Mathematics}. \newblock AK Peters, 2004. \bibitem{FB} H. R. P. Ferguson and D. H. Bailey. \newblock A polynomial time, numerically stable integer relation algorithm. \newblock RNR Techn. Rept. RNR-91-032, Jul. 14, 1992. \bibitem{Plouffe} Plouffe,~S. \newblock Identities inspired by Ramanujan notebooks (part 2), \newblock \href{http://www.plouffe.fr/simon/}{\tt http://www.plouffe.fr/simon/}, April 2006. \bibitem{Vepstas} Vepstas,~L. \newblock On Plouffe's Ramanujan identities, \newblock \href{http://arxiv.org/abs/math/0609775}{\tt http://arxiv.org/abs/math/0609775}, November 2010. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 11Y60. \noindent \emph{Keywords: } $\pi$, Riemann zeta function. \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received January 24 2011; revised version received February 7 2011. Published in {\it Journal of Integer Sequences}, February 20 2011. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}