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

2022/10/12 by Zhu, Xiao-Jie
#11F11 #11F20 #33B15 #FOS: Mathematics #Number Theory (math.NT) #Primary 11F30 #Secondary 11F03

paper · doi:10.48550/arxiv.2210.06253

Abstract

We give all possible holomorphic Eisenstein series on Γ₀(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.

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