The Riemann zeta in terms of the dilogarithm

2011/01/25 by Sergio Albeverio, Albeverio, Sergio, Claudio Cacciapuoti +1
Mathematics · #11F27 (Primary) 42A38 (Secondary) #11M06 #11M26 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F27 #msc:11M06 #msc:11M26 #msc:42A38

paper · pdf · doi:10.48550/arxiv.1101.4786

revised version, major changes in Sec. 3 and 5, 25 pages

arxiv created 2012/08/13 · arxiv updated 2012/08/14

Abstract

We give a representation of the classical Riemann ζ-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 Gl₂-function). We also derive corresponding representations involving the derivatives of the Gl₂-function. A generalized symmetrized Müntz-type formula is also derived. For a special choice of test functions it connects to our integral representation of the ζ-function, providing also a computation of a concrete Mellin transform. Certain formulae involving series of zeta functions and gamma functions are also derived.

Related

• On Müntz-type formulas related to the Riemann zeta function
• The Mellin transform of the square of Riemann's zeta-function
• On certain Fourier expansions for the Riemann zeta function
• A Finite Reflection Formula For A Polynomial Approximation To The Riemann Zeta Function
• Summation rules for the values of the Riemann zeta-function and generalized harmonic numbers obtained using Laurent developments of polygamma functions and their products