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My question is: What are the periodic Dirichlet series? Does the Riemann zeta function $ζ(s)=\sum_{n=1}^\infty \frac{1}{n^s}$ and the alternating zeta function $η(s)=\sum_{n=1}^\infty (-1)ⁿ⁻¹/n^{s}$ are periodic?

DER
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1 Answers1

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Look in that thesis by Tapas Chatterjee, where you find the answer is yes for both the Riemann zeta function and the alternating zeta function.

user64494
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    Why would you link to a PhD thesis to answer this? – Pedro Sep 29 '13 at 17:26
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    @ user64494: Since you read this thesis, can you indicate to me the page where this fact was presented. – DER Sep 29 '13 at 17:27
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    @AR1 : p.21, lines 6-8. – user64494 Sep 29 '13 at 17:40
  • @ user64494: Thank you very much. – DER Sep 29 '13 at 17:41
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    @ Pedro Tamaroff: That is easily accessible through the Internet. – user64494 Sep 29 '13 at 17:42
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    You didn't answer to the question : what is a periodic Dirichlet series. Here it means the coefficients are $q$-periodic so $$\Gamma(s) \sum_{n=1}^\infty a_n n^{-s} = \int_0^\infty x^{s-1} \frac{\sum_{n=1}^q a_n e^{-nx}}{1-e^{-qx}}dx$$ also giving a functional equation in term of $\sum_{m=1}^q a_m \sum_{n=1}^\infty e^{2i \pi m n/ q} n^{-s}$ – reuns Jul 07 '17 at 14:55