A two variable 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...

Two-Variable Zeta-Functions on Graphs and Riemann–Roch Theorems

We investigate, in this article, a generalization of the Riemann–Roch theorem for graphs of Baker and Norine, with a view toward identifying new objects for which a two-variable zeta-function can be defined. To a lattice Λ of rank n − 1 in Z n and perpendicular to a positive integer vector R, we define the notions of g-number and of canonical vector , in analogy with the notions of genus and ca...

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

On a Two-Variable Zeta Function for Number Fields

This paper studies a two-variable zeta function ZK(w, s) attached to an algebraic number field K, introduced by van der Geer and Schoof [11], which is based on an analogue of the RiemannRoch theorem for number fields using Arakelov divisors. When w = 1 this function becomes the completed Dedekind zeta function ζ̂K(s) of the field K. The function is an meromorphic function of two complex variable...