The Riemann zeta in terms of the dilogarithm

作者：Albeverio Sergio; Cacciapuoti Claudio^*

来源：Journal of Number Theory, 2013, 133(1): 242-277.

DOI：10.1016/j.jnt.2012.06.002

• Classical Riemann%26apos
• s zeta function
• Dilogarithm function
• Mellin transform
• Meromorphic character of zeta
• Zero-free regions of the zeta function
• Bounds on the zeta function
• Integral representations of the zeta function
• Series of zeta functions
• Series of gamma functions
• Generalized symmetrized Muntz formula

We give a representation of the classical Riemann zeta-function in the half plane Res %26gt; 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 Muntz-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.

• 出版日期2013-1