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

Xiao-Jie Zhu

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arXiv · 2210.06253

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

Xiao-Jie Zhu

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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.

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Xiao-Jie Zhu. 2022-10-12. Holomorphic Eisenstein series of rational weights and special values of Gamma function. https://arxiv.org/abs/2210.06253

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