A continuous version of multiple zeta values with double variables

TL;DR

This paper introduces a continuous version of multiple zeta functions with double variables, explores their algebraic properties, including shuffle products, and demonstrates their relation to cyclotomic multiple zeta values, with applications to Ramanujan's identities.

Contribution

It defines continuous multiple zeta values with double variables, proves their algebraic properties, and establishes their inclusion of cyclotomic multiple zeta values of level 2.

Findings

Continuous multiple zeta values satisfy shuffle product relations.

Finite-dimensionality of the space generated by these values.

Connection to cyclotomic multiple zeta values of level 2.

Abstract

In this paper we define a continuous version of multiple zeta functions with double variables. They can be analytically continued to meromorphic functions on $\mathbb{C}^r$ with only simple poles at some special hyperplanes. The evaluations of these functions at positive integers (continuous multiple zeta values) satisfy the first shuffle product and the second shuffle product. We proved that the dimension of the $\mathbb{Q}-$linear spaces generated by continuous multiple zeta values with given weight are finite. By using a theorem of C.Glanois, we proved that continuous multiple zeta values include all cyclotomic multiple zeta values of level 2. We will give a detail analysis about the two different shuffle products. Furthermore, we will discuss the extension of the two different products, we proved a theorem about comparing the two different shuffle product, this is an analogy of Ihara-Kaneko-Zagier's comparison theorem in the case of continuous multiple zeta values. As an application, we will give a new method to proof some Ramanujan's identities. Finally, we will provide some conjectures.