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

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.

arXiv comment27 pages, accepted for publication in Acta Arithmetica