## Abstract

An integral representation of the $p$-series of odd $p$ is shown;$$\sum _{n=1}^{\mathrm{\infty}}\frac{1}{{n}^{2p+1}}=(-1{)}^{p}\frac{(2\pi {)}^{2p}}{\left(2p\right)!}{\int}_{0}^{1}{B}_{2p}(t\left)\mathrm{log}\right(\mathrm{sin}\pi t\left)\mathrm{d}t\phantom{\rule{1em}{0ex}}\right(p=1,2,\dots ),$$where ${B}_{2p}\left(t\right)$ is a Bernoulli polynomial of degree $2p$. As a consequence of this we have $$\sum _{n=1}^{\mathrm{\infty}}\frac{1}{{n}^{2p+1}}=(-1{)}^{p}\frac{(2\pi {)}^{2p}}{\left(2p\right)!}2\left[\sum _{k=0}^{p}\left(\begin{array}{c}2p\\ 2k\end{array}\right){B}_{2p-2k}\left(\frac{1}{2}\right){b}_{2k}\right],$$where ${b}_{2k}={\int}_{0}^{\frac{1}{2}}{t}^{2k}\mathrm{log}\left(\mathrm{cos}\pi t\right)\mathrm{d}t$, $k=0,1,\dots ,p$.

## 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

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