Extended Riemann Zeta Functions

Multiple finite Riemann zeta functions

Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some q-series identity for proving the zeta function has an Euler product and then, describe the location of zeros. We study further multi-variable and multi-parameter versions of the multiple finite Riemann zeta functions and their infinite cou...

The Riemann Zeta and Allied Functions

Zeta functions that hear the shape of a Riemann surface

To a compact hyperbolic Riemann surface, we associate a finitely summable spectral triple whose underlying topological space is the limit set of a corresponding Schottky group, and whose “Riemannian” aspect (Hilbert space and Dirac operator) encode the boundary action through its Patterson-Sullivan measure. We prove that the ergodic rigidity theorem for this boundary action implies that the zet...

Complex Zeros of Two Incomplete Riemann Zeta Functions

The computation of the complex zeros of an incomplete Riemann zeta function defined in an earlier paper is extended and new zero trajectories are given. A second incomplete Riemann zeta function is denned and its zero trajectories are investigated numerically as functions of the upper limit X of the definition integral. It becomes apparent that there exist three different classes of zero trajec...

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...