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Jeffery Lagarias, in his recent article Euler's constant: Euler's work and modern developments in the AMS Bulletin, mentions that Euler obtained $\zeta(3)={{2\pi^3 b(3/2)}\over 3}$ for some "Bernoulli function" $b(z)$ in Section 2.4 ("Zeta values") of his article.

Lagarias also mentions that Euler similarly obtained a value for $\zeta(5)$ in terms of $b(5/2)$.

Is there any modern article explaining this "Bernoulli function $b(z)$?

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I found something on Wikipedia: https://en.wikipedia.org/wiki/Bernoulli_number#Integral_representation_and_continuation It is written that $$ b\left( s \right) = 2e^{\frac{\pi }{2}is} \int_0^{ + \infty } {\frac{{st^s }}{{1 - e^{2\pi t} }}\frac{{dt}}{t}} $$ and $b\left( {2n} \right) = B_{2n}$ for any $n>0$, where $B_k$ is the $k$th Bernoulli number. Also $$ \zeta \left( 3 \right) = \frac{{2\pi ^3 b\left( 3 \right)}}{{3i}},\quad \zeta \left( 5 \right) = \frac{{2i\pi ^5 b\left( 5 \right)}}{{15}} . $$

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Maybe this paper will be helpful: http://arxiv.org/abs/1005.2733 (Donal F. Connon, A generalisation of the Bernoulli numbers from the discrete to the continuous).

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