From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: alternating series
Date: Fri, 03 Dec 1999 19:45:47 GMT
Newsgroups: sci.math
Keywords: values of certain Dirichlet L-functions

In article <828n0n$8im$1@nnrp1.deja.com>,
  Serge Zlobin <sirg@my-deja.com> wrote:
>   Hi,
> Long ago I had recieved the following result:
> 1/1^3-1/2^3+1/4^3-1/5^3+1/7^3-1/8^3+...=4*sqrt(3)*(pi^3)/243.
> But my proof was sophisticated and difficult. Does anybody know how to
> calculate sums of that type?
>   Here is some information that had been gotten after the discussion in
> fido7.ru.math.
>   Consider the sums s(k)=1/1^k-1/2^k+1/4^k-1/5^k+1/7^k-1/8^k+...,
> k is odd.
>   Then s(1)=pi*sqrt(3)/9, s(3)=4*sqrt(3)*(pi^3)/243.
>   Through expiremental evalution it was established, that
> s(n)=a(n)*(2*pi)^n / (3^n*sqrt(3)*n!),
> where a(1)=1/2, a(3)=1, a(5)=5, a(7)=49, a(9)=809, a(11)=20317,
> a(13)=722813. It seems to be true that a(n) is odd integer (when n>1)
> and n divides a(n).
>   Searching the sequence 1,5,49,809,20317,722813 brings to the reference
> about Glaisher's G numbers.

This is asking for the values at odd positive integers of the
Dirichlet L-function with conductor 3.

Let L(s,chi) = sum_{n=1}^infinity chi(n)/n^s
where chi(n) = 0, 1 or -1 according to if n = 0, 1 or 2 mod 3.
Then L(k, chi) is a rational multiple of pi^k when k is odd.
(This is a general property of Dirchlet L-functions, their
values at positive integers k with the same parity as the conductor
are rational multiples of pi^k).

One can evaluate these values by adapting the usual proof
of the zeta-function evaluation at even integers. Recall that
pi cot(pi z) = 1/z + sum_{m=1}^infinity [1/(z + m) + 1/(z - m)].
Replace z by (z - 1/3) and expanding out each summand as a geometric
series we get that the z^{2r} coefficient in the Macluarin series
of pi cot(pi (z-1/3)) is a multiple of L(2r+1, chi). Presumably
the Glaisher numbers occur in the Maclaurin series of
this function.

--
Robin Chapman
 http://www.maths.ex.ac.uk/~rjc/rjc.html
  "`Well, I'd already done a PhD in X-Files Theory at UCLA, ...'"
    Greg Egan, _Teranesia_


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