The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a+b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characte...

The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on (0,?), which equivalent to Riemann hypothesis. This involves dilations fractional part function by factors ? k ?(0,1), k?1. We develop probabilistic extensions considering these as random: this yields new structures and criteria, one them having a significant overlap with general strong Báez-...

In the paper, it is obtained that there are infinite discrete shifts Ξ(s+ikh), h>0, k∈N0 of Mellin transform Ξ(s) square Riemann zeta-function, approximating a certain class analytic functions. For proof, probabilistic approach based on weak convergence probability measures in space functions applied.

Using Fourier analysis, we study local limit theorems in weak-convergence problems. Among many applications, we discuss random matrix theory, some probabilistic models in number theory, the winding number of complex brownian motion and the classical situation of the central limit theorem, and a conjecture concerning the distribution of values of the Riemann zeta function on the critical line.

This paper studies the interplay between probability, number theory, and geometry in the context of relatively prime integers in the ring of integers of a number field. In particular, probabilistic ideas are coupled together with integer lattices and the theory of zeta functions over number fields in order to show that P (gcd(z1, z2) = 1) = 1 ζQ(i)(2) where z1, z2 ∈ Z[i] are randomly chosen and...

Abstract. The series expansion of a power of the modified Bessel function of the first kind is studied. This expansion involves a family of polynomials introduced by C. Bender et al. New results on these polynomials established here include recurrences in terms of Bell polynomials evaluated at values of the Bessel zeta function. A probabilistic version of an identity of Euler yields additional ...

For a class of subordinators we investigate the spectrum of the infinitesimal generator of subordinate Brownian motion on a closed manifold. We consider three spectral functions of the generator: the zeta function, the heat trace and the spectral action. Each spectral function explicitly yields both probabilistic and geometric information, the latter through the classical heat invariants. All c...