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 → (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 ↦ 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.
- Publication:
-
Journal of Physics A Mathematical General
- Pub Date:
- June 2020
- DOI:
- 10.1088/1751-8121/ab8d51
- arXiv:
- arXiv:2004.12723
- Bibcode:
- 2020JPhA...53w5205S
- Keywords:
-
- Riemann zeta-function;
- functional equations;
- diffusion processes;
- Mathematical Physics
- E-Print:
- Version to match the one to appear in Journal of Physics A