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.

arXiv commentVersion to match the one to appear in Journal of Physics A

References in corpus (11)

• Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem
• Physics of the Riemann Hypothesis
• Quantum mechanical potentials related to the prime numbers and Riemann zeros
• Duality, Partial Supersymmetry, and Arithmetic Number Theory
• Quantum hamiltonians and prime numbers
• The Riemann zeros as spectrum and the Riemann hypothesis
• Zeta Function Zeros, Powers of Primes, and Quantum Chaos
• Riemann zeta zeros and prime number spectra in quantum field theory
• Supersymmetry, Vacuum Statistics, and the Fundamental Theorem of Algebra
• Riemann zeta zeros and zero-point energy
• One-loop effective action and the Riemann Zeros