The Riemann zeta in terms of the dilogarithm

Details The Riemann zeta in terms of the dilogarithm The Riemann zeta in terms of the dilogarithm

2011-01-25

arxiv.org/abs/1101.4786v1

Mathematics

Number Theory

28 pages

Scientific paper

We give a representation of the classical Riemann $\zeta$-function 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. As corollaries we derive new proofs of zero-free regions for the $\zeta$-function and of bounds on it. A generalized symmetrized M\"untz-type formula is derived. For a special choice of test functions it gives a new derivation of the functional equation and provides a computation of a Mellin transform. Certain formulas involving series of zeta functions are also derived.

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