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I am aware of the generalized Bernoulli numbers, but these are not what I'm looking for. I was wondering if there exists such a thing as fractional, real or even complex Bernoulli numbers ( $B_z$ for $z \in \mathbb{C}$).

My motivation comes from the Ramanujan Summation, as the Bernoulli numbers are involved in it. I was hoping that, if the Bernoulli numbers could be extendend, so could perhaps the Ramanujan Summation, allowing it to assign a sum to a wider class of divergent series.

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    Do you think there is a chance that Ramanujan summation could be extended by means of some ideas that are loosely based on the things I just said in the comments?2011-06-08

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Ramanujan himself gave a definition of Bernoulli numbers for a complex index - see Ramanujan's Notebooks - Part 1 by Bruce C. Berndt, Chapter 5 equation (25.1) with further results in Chapter 7.


Added by J. M.:

Ramanujan's definition of the Bernoulli numbers, as given in Berndt's book, is

$B_s^\ast=\frac{2\Gamma(s+1)}{(2\pi)^s}\zeta(s)$

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    This is a nice precise answer, thanks!2011-06-08