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\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}

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\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
$$
\frac{x}{e^x - 1} = \sum_{n=0}^\infty \frac{B_n x^n}{n!}, ~~~|x|&lt;2\pi.
$$
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 &amp; = &amp; 72 S_1 (1) - 96 S_1 (2) + 24 S_1 (4) \\
\pi^3 &amp; = &amp; 720 S_3 (1) - 900 S_3 (2) + 180 S_3 (4) \\
\pi^5 &amp; = &amp; 7056 S_5 (1) - 6993 S_5 (2) + 63 S_5 (4) \\
\pi^7 &amp; = &amp; \frac{907200}{13} S_7 (1)-70875 S_7 (2)+\frac{14175}{13} S_5 (4)
\end{eqnarray*}
and
\begin{eqnarray*}
\zeta (3) &amp; = &amp; 28 S_3 (1) - 37 S_3 (2) + 7 S_3 (4) \\
\zeta (5) &amp; = &amp; 24 S_5 (1) - \frac{259}{10} S_5 (2) + \frac{1}{10} S_5 (4) \\
\zeta (7) &amp; = &amp; \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}
&gt; with(IntegerRelations):
&gt; Digits := 100;
&gt; S := r -&gt; sum( 1/k/( exp(Pi*r*k)-1 ), k=1..infinity );
&gt; 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 $\alpha&gt;0$, $\beta&gt;0$, 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\} =  }\\
&amp; &amp; 
(-\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)= } \\
&amp; &amp; 
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)
= }\\
&amp; &amp; 
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) &amp; = &amp;
-\frac{ 16^m( G(2m)16^m+2E_m) }{(-1+16^m)E_m} S_{4m+1}(\pi) \\
&amp; &amp;
-\frac{ 2G_{2m}K_m 16^m (2\cdot 16^m - (-4)^m + 1) }
  {(-1+16^m)E_m} S_{4m+1}(2\pi) \\ 
&amp; &amp; 
-\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*}



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\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}



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\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.

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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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\end{document}

                                                                                

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