有理权的全纯爱森斯坦级数和函数的特殊值

IF 0.5 3区数学 Q3 MATHEMATICS

Acta Arithmetica Pub Date : 2022-10-12 DOI:10.4064/aa221110-1-4

Xiaojie Zhu

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引用次数: 0

摘要

我们给出了$\Gamma_0（p）$上所有可能的全纯Eisenstein级数，其有理权大于$2$，并且乘子系统在所有尖端与某些有理权η商相同。我们证明了它们是模形式，并给出了它们的傅立叶展开式。我们建立了四类恒等式，将这些级数等同于有理权η商。作为一个应用，我们给出了Euler-Gamma函数在任意有理参数下的特殊值的级数表达式。这些表达式涉及Dedekind和的指数和。

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

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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