WEIGHTED DISCRETE UNIVERSALITY OF THE RIEMANN ZETA-FUNCTION

An Effective Universality Theorem for the Riemann Zeta Function

Let 0 < r < 1/4, and f be a non-vanishing continuous function in |z| ≤ r, that is analytic in the interior. Voronin’s universality theorem asserts that translates of the Riemann zeta function ζ(3/4 + z + it) can approximate f uniformly in |z| < r to any given precision ε, and moreover that the set of such t ∈ [0, T ] has measure at least c(ε)T for some c(ε) > 0, once T is large enough. This was...

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...

Mollifying the Riemann Zeta-function

Aspects of Analytic Number Theory: The Universality of the Riemann Zeta-Function

Abstract. These notes deal with Voronin’s universality theorem which states, roughly speaking, that any non-vanishing analytic function can be uniformly approximated by certain shifts of the Riemann zeta-function. We start with a brief introduction to the classical theory of the zeta-function. Then we give a self-contained proof of the universality theorem. We conclude with several interesting ...

Path Integrals and Voronin’s Theorem on the Universality of the Riemann Zeta Function

This talk is based on work with N. Khuri and H. Ren.[1,2] The main new tools we use are theorems by Voronin on the universality properties of the Riemann Zeta function, ζ(s), in the critical strip, 1 2 < Re s < 1. Given any real continuous function, φ(x), 0 ≤ x ≤ L, we can choose a mapping, s(x), which maps the line, 0 ≤ x ≤ L, onto a line s(x) that lies in the strip 1 2 < Re s < 1, then given ...