q-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...

On q - analogues of Riemann ’ s zeta

The aim of the paper is to define q-deformations of the Riemann zeta function and to extend them to the whole complex plane. The construction is directly related to the recent difference generalization of the Harish-Chandra theory of zonal spherical functions [C1,C2,C3]. The Macdonald truncated theta function [M1] replaces xs. The analytic continuation is based on the shift operator technique a...

q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function

A q-analogue ζq(s) of the Riemann zeta function ζ(s) was studied in [Kaneko et al. 03] via a certain q-series of two variables. We introduce in a similar way a q-analogue of the Dirichlet L-functions and make a detailed study of them, including some issues concerning the classical limit of ζq(s) left open in [Kaneko et al. 03]. We also examine a “crystal” limit (i.e. q ↓ 0) behavior of ζq(s). T...

Lagrangians with Riemann Zeta Function

We consider construction of some Lagrangians which contain the Riemann zeta function. The starting point in their construction is p-adic string theory. These Lagrangians describe some nonlocal and nonpolynomial scalar field models, where nonlocality is controlled by the operator valued Riemann zeta function. The main motivation for this research is intention to find an effective Lagrangian for ...

Mollifying the Riemann Zeta-function