On an integral representation of special values of the zeta function at odd integers

Abstract

An integral representation of the p-series of odd p is shown;
$¥sum^¥infty_{n=1} ¥frac{1}{n^{2p+1}}$ = (-1)^p$¥frac{(2¥pi)^{2p}}{(2p)!} ¥int^1_0$B_2p(t) log(sinπt)dt (p=1,2,…),
where B_2p(t) is a Bernoulli polynomial of degree 2p. As a consequence of this we have
$¥sum^¥infty_{n=1} ¥frac{1}{n^{2p+1}}$ = (-1)^p$¥frac{(2¥pi)^{2p}}{(2p)!} 2 ¥bigg[ ¥sum^p_{k=0} ¥bigg( {2p ¥atop 2k} ¥bigg) B_{2p-2k} ¥bigg( ¥frac12 ¥bigg) b_{2k} ¥bigg],$
where b_2k = ∫^$¥frac12$₀ t^2k log(cosπt)dt, k = 0,1,…,p.