Hidden Markov Processes (HMP) is one of the basic tools of the modern probabilistic modeling. The characterization of their entropy remains however an open problem. Here the entropy of HMP is calculated via the cycle expansion of the zeta-function, a method adopted from the theory of dynamical systems. For a class of HMP this method produces exact results both for the entropy and the moment-gen...

Two representations of the Bessel zeta function are investigated. An incomplete representation is constructed using contour integration and an integral due to Hawkins fully evaluated (analytically continued) produce two infinite series. This new representation, at integer values argument, produces results that consistent with known (values, slope, pole structure). Not surprisingly, studied foun...

Let the summatory function of the Möbius function be denoted M(x). We deduce in this article conditional results concerning M(x) assuming the Riemann Hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the Riemann zeta function. The main results shown are that the weak Mertens conjecture and the existence of a limiting distribution of e−y/2M(ey) are consequences of the af...

We introduce a theory of probabilistic renormalization for series, the renormalized values being encoded in expectation certain random variable on set natural numbers. identify large class weakly renormalizable series Dirichlet type, whose analysis depends properties (infinite order) difference operator that we call Bernoulli operator. For this class, show is compatible with analytic continuati...

In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by second author. Applying Mellin transformation to function, define new class zeta type functions, which is related interpolation functions Apostol–Bernoulli polynomials, Bernoulli Euler polynomials. This Hurwitz alternating Lerch function. Furthermore, using these derive ...