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

Volume 210 / 2023

Xiao-Jie Zhu Acta Arithmetica 210 (2023), 279-305 MSC: Primary 11F30; Secondary 11F03, 11F11, 11F20, 33B15. DOI: 10.4064/aa221110-1-4 Published online: 23 August 2023


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 the Euler Gamma function at any rational arguments. These expressions involve exponential sums of Dedekind sums.


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