Functional equations for regularized zeta-functions and diffusion processes

Abstract

We discuss modifications in the integral representation of the Riemann zeta-function that lead to generalizations of the Riemann functional equation that preserves the symmetry $s\to (1-s)$ in the critical strip. By modifying one integral representation of the zeta-function with a cut-off that does exhibit the symmetry $x\mapsto 1/x$, we obtain a generalized functional equation involving Bessel functions of second kind. Next, with another cut-off that does exhibit the same symmetry, we obtain a generalization for the functional equation involving only one Bessel function of second kind. Some connection between one regularized zeta-function and the Laplace transform of the heat kernel for the Euclidean and hyperbolic space is discussed.

Version to match the one to appear in Journal of Physics A

References in corpus (3)

• Physics of the Riemann Hypothesis
• Quantum mechanical potentials related to the prime numbers and Riemann zeros
• One-loop effective action and the Riemann Zeros