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On an integral representation of special values of the zeta function at odd integers. (English) Zbl 1102.11014

The author derives for the Riemann zeta function \(\zeta(s)\) the formula \[ \zeta(2p+1) = (-1)^p{{(2\pi)^{2p}}\over{(2p)!}}\int_0^1 B_{2p}(t)\log(\sin \pi t)\,dt \quad (p=1,2,\dots), \] where \(B_{2p}(t)\) is the Bernoulli polynomial of degree \(2p\). He uses the functions \(\widetilde{B}_p(x)\) defined on the real line as the periodic functions with period 1 so that \(\widetilde{B}_p(x)=B_p(x)\) for \(0 \leq x < 1\). The result is obtained on considering the Fourier series expansions and the convolutions of these functions.
Another interesting expression for \(\zeta(2p+1)\) follows from the observation that the integral in the formula above equals \(2\sum_{k=0}^p {{2p}\choose{2k}}B_{2p-2k}({1\over 2})b_{2k}\), where \(b_{2k}\) is the \(2k\)th moment of \(\log(\cos \pi t)\) over the interval \([0,\,{1\over 2}]\).

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
42A85 Convolution, factorization for one variable harmonic analysis
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References:

[1] L. Euler, Exercitationes Analyticae, Novi Commentarii Academiae Scientiarum Petropolitanae, 17 (1772), 173-204. Collected Works, I-15, pp.,131-167.
[2] S. Kurokawa, M. Wakayama and T. Momotani, Translators’ Commentaries, In: W. Dunham, Euler, The Master of Us All, Springer-Verlag, Tokyo, 2004, pp.,231-250.
[3] T. Takagi, An Introduction to Analaysis, 3rd Ed., Iwanami Shoten, Tokyo 1961.
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