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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Exponential sums of Lerch's zeta functions


Author: Kai Wang
Journal: Proc. Amer. Math. Soc. 95 (1985), 11-15
MSC: Primary 11M35
DOI: https://doi.org/10.1090/S0002-9939-1985-0796438-5
MathSciNet review: 796438
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Abstract: For $ x$ not an an integer and $ \operatorname{Re} (s) > 0$, let

$\displaystyle F(x,s) = \sum\limits_{k = 1}^\infty {\frac{{{e^{2\pi ikx}}}}{{{k^s}}}} $

be the Lerch's zeta function. In this note, we will show that

$\displaystyle \sum\limits_{\gamma = 1}^{m - 1} {{e^{ - 2\pi i\gamma \alpha /m}}... ...left( {\frac{\alpha }{m} - \left[ {\frac{\alpha }{m}} \right]} \right)} \right)$

where $ \alpha $ is an integer and $ \alpha \not\equiv 0(\mod m)$ and $ n \geqslant 1$. For $ n = 1$, this formula is equivalent to the classical Eisenstein formula

$\displaystyle \frac{\alpha }{m} - \left[ {\frac{\alpha }{m}} \right] - \frac{1}... ...1}^{m - 1} {\sin \frac{{2\pi \gamma \alpha }}{m}} \cot \frac{{\pi \gamma }}{n}.$


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DOI: https://doi.org/10.1090/S0002-9939-1985-0796438-5
Article copyright: © Copyright 1985 American Mathematical Society