Markov measures determine the zeta function

The Probabilistic Zeta Function

This paper is a summary of results on the PG(s) function, which is the reciprocal of the probabilistic zeta function for finite groups. This function gives the probability that s randomly chosen elements generate a group G, and information about the structure of the group G is embedded in it.

The Fibonacci Zeta Function

We consider the lacunary Dirichlet series obtained by taking the reciprocals of the s-th powers of the Fibonacci numbers. This series admits an analytic continuation to the entire complex plane. Its special values at integral arguments are then studied. If the argument is a negative integer, the value is algebraic. If the argument is a positive even integer, the value is transcendental by Neste...

Mollifying the Riemann Zeta-function

q-Riemann zeta function

We consider the modified q-analogue of Riemann zeta function which is defined by ζq(s)= ∑∞ n=1(qn(s−1)/[n]s), 0< q < 1, s ∈ C. In this paper, we give q-Bernoulli numbers which can be viewed as interpolation of the above q-analogue of Riemann zeta function at negative integers in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Also, we will treat some...

A limit theorem for Bohr–Jessen’s probability measures of the Riemann zeta-function

The asymptotic behavior of value distribution of the Riemann zeta-function ζ(s) is determined for 1 2 < (s) < 1. Namely, the existence is proved, and the value is given, of the limit lim →∞ ( (log )σ)−1/(1−σ) logW (C \R( ), σ, ζ) for 1 2 < σ < 1, where R( ) is a square in the complex plane C of side length 2 centered at 0, and W (A,σ, ζ) = lim T→∞ (2T )−1μ1({t ∈ [−T, T ] | log ζ(σ + t √−1) ∈ A}...