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Proceedings of the American Mathematical Society

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Two new proofs of Lerch's functional equation

Author: Bruce C. Berndt
Journal: Proc. Amer. Math. Soc. 32 (1972), 403-408
MSC: Primary 10H05
MathSciNet review: 0297721
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Abstract: One bright Sunday morning I went to church, And there I met a man named Lerch. We both did sing in jubilation, For he did show me a new equation.

Two simple derivations of the functional equation of

$\displaystyle \sum\limits_{n = 0}^\infty {\exp [2\pi inx]{{(n + a)}^{ - s}}} $

are given. The original proof is due to Lerch.

References [Enhancements On Off] (What's this?)

  • [1] M. Lerch, Note sur la fonction $ \Re (w,x,s) = \sum\nolimits_{k = 0}^\infty {{e^{2k\pi ix}}/{{(w + k)}^s}} $, Acta Math. 11 (1887), 19-24. MR 1554747
  • [2] E. C. Titchmarsh, The theory of functions, 2nd ed., Oxford Univ. Press, London, 1939. MR 0197687 (33:5850)

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Keywords: Lerch's zeta-function, Hurwitz zeta-function, Riemann zeta-function, functional equation, Euler-Maclaurin summation formula
Article copyright: © Copyright 1972 American Mathematical Society

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