A note on odd zeta values over any number field and Extended Eisenstein 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${}_2(\mathbb{Z})$. 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.