A note on odd zeta values over any number field and Extended Eisenstein series

TL;DR

This paper generalizes Ramanujan's identity for the Riemann zeta function to odd zeta values over any number field, introduces a new Eisenstein series extension, and explores their transformation properties and applications.

Contribution

It introduces a transformation formula for odd zeta values over arbitrary number fields and constructs an extended Eisenstein series with modular transformation properties.

Findings

Generalized Ramanujan's identity for number fields

Constructed a new Eisenstein series with modular transformation

Provided applications in studying zeta values and Lambert series

Abstract

In this article, we have studied transformation formulas of zeta function at odd integers over an arbitrary number field which in turn generalizes Ramanujan's identity for the Riemann zeta function. The above transformation leads to a new number field extension of Eisenstein series, which satisfies the transformation $z \mapsto -1/z$ like an integral weight modular form over SL$_2(\Z)$. The results provide number of important applications, which are important in studying the behaviour of odd zeta values as well as Lambert series in an arbitrary number field.