I read somewhere that there exists a zeta function for Gaussian integers that involves a combination of the Riemann zeta function and the Dirichlet beta function. Unfortunately, I have lost my source and and searching on internet on this subject does not yield the desired results. Can anyone give me such a definition for Gaussian integers, and possibly some articles I can read on the subject?
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3See p.6 of https://www.maths.nottingham.ac.uk/personal/cw/download/fnt_chap5.pdf , and look up the Dirichlet $L$-function, which actually simplifies into another zeta in this case. – Chappers Jul 27 '17 at 10:42
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See this or this threads (and links) which may help too, – Raymond Manzoni Jul 27 '17 at 10:51
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For any number field there is a certain Dirichlet series called a Dedekind Zeta Function. The Dedekind Zeta function for $\Bbb{Q}[i]$ "factors" as the Riemann Zeta function and Dirichlet Beta function. – sharding4 Jul 27 '17 at 12:14
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Or thisone and see a course on the splitting of prime ideals in $\mathbb{Q}(i)$. – reuns Jul 27 '17 at 15:08