Open Access
Translator Disclaimer
July, 2006 On an integral representation of special values of the zeta function at odd integers
Takashi ITO
J. Math. Soc. Japan 58(3): 681-691 (July, 2006). DOI: 10.2969/jmsj/1156342033

Abstract

An integral representation of the p -series of odd p is shown; n = 1 1 n 2 p + 1 = ( - 1 ) p ( 2 π ) 2 p ( 2 p ) ! 0 1 B 2 p ( t ) log ( sin π t ) d t ( p = 1 , 2 , ) , where B 2 p ( t ) is a Bernoulli polynomial of degree 2 p . As a consequence of this we have n = 1 1 n 2 p + 1 = ( - 1 ) p ( 2 π ) 2 p ( 2 p ) ! 2 k = 0 p 2 p 2 k B 2 p - 2 k 1 2 b 2 k , where b 2 k = 0 1 2 t 2 k log ( cos π t ) d t , k = 0 , 1 , , p .

Citation

Download Citation

Takashi ITO. "On an integral representation of special values of the zeta function at odd integers." J. Math. Soc. Japan 58 (3) 681 - 691, July, 2006. https://doi.org/10.2969/jmsj/1156342033

Information

Published: July, 2006
First available in Project Euclid: 23 August 2006

zbMATH: 1102.11014
MathSciNet: MR2254406
Digital Object Identifier: 10.2969/jmsj/1156342033

Subjects:
Primary: 11B68
Secondary: 42A85‎

Keywords: Bernoulli polynomials , Convolutions , Fourier series

Rights: Copyright © 2006 Mathematical Society of Japan

JOURNAL ARTICLE
11 PAGES


SHARE
Vol.58 • No. 3 • July, 2006
Back to Top