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arXiv · 1101.4786

The Riemann zeta in terms of the dilogarithm

Sergio Albeverio·Claudio Cacciapuoti
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Abstract

We give a representation of the classical Riemann $\zeta$-function in the half plane $\Re s>0$ in terms of a Mellin transform involving the real part of the dilogarithm function with an argument on the unit circle (associated Clausen $Gl_2$-function). We also derive corresponding representations involving the derivatives of the $Gl_2$-function. A generalized symmetrized M\"untz-type formula is also derived. For a special choice of test functions it connects to our integral representation of the $\zeta$-function, providing also a computation of a concrete Mellin transform. Certain formulae involving series of zeta functions and gamma functions are also derived.

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Sergio Albeverio, Claudio Cacciapuoti. 2011-01-25. The Riemann zeta in terms of the dilogarithm. https://arxiv.org/abs/1101.4786

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