Holomorphic Eisenstein series of rational weights and special values of Gamma function

TL;DR

This paper classifies holomorphic Eisenstein series of rational weights on a0a0(p), proves their modularity, and connects them to special values of the Gamma function through identities involving eta-quotients and exponential sums.

Contribution

It provides a complete classification of certain rational-weight Eisenstein series and links them to Gamma function values using novel identities and Fourier expansions.

Findings

Classified all such Eisenstein series on a0a0(p)

Proved these series are modular forms with explicit Fourier expansions

Derived series expressions for Gamma function values at rational points

Abstract

We give all possible holomorphic Eisenstein series on $\Gamma_0(p)$, of rational weights greater than $2$, and with multiplier systems the same as certain rational-weight eta-quotients at all cusps. We prove they are modular forms and give their Fourier expansions. We establish four sorts of identities that equate such series to rational-weight eta-quotients. As an application, we give series expressions of special values of Euler Gamma function at any rational arguments. These expressions involve exponential sums of Dedekind sums.