Riemann surface of the Riemann zeta function

TL;DR

This paper extends the classical Riemann zeta function to a three-variable setting involving infinite-dimensional complex sequences and studies its meromorphic continuation.

Contribution

It introduces a novel three-variable formulation of the Riemann zeta function and analyzes its meromorphic continuation in this broader context.

Findings

Established the meromorphic continuation of the three-variable zeta function.

Connected the new formulation to the classical Riemann zeta function.

Provided insights into the structure of the zeta function in infinite-dimensional settings.

Abstract

In this paper we treat the classical Riemann zeta function as a function of three variables: one is the usual complex $\adyn$-dimensional, customly denoted as $s$, another two are complex infinite dimensional, we denote it as $\b = \{b_n\}_{n=1}^{\infty}$ and $\z =\{z_n\}_{n=1}^{\infty}$. When $\b = \{1\}_{n=1}^{\infty}$ and $\z = \{\frac{1}{n}\}_{n=1}^{\infty}$ one gets the usual Riemann zeta function. Our goal in this paper is to study the meromorphic continuation of $\zeta (\b , \z ,s)$ as a function of the triple $(\a , \z , s)$. Minor corrections, to appear in the Journal of Mathematical Analysis and Applications.