Explicit identities on zeta values over imaginary quadratic field

2021/05/10 by Soumyarup Banerjee, Rahul Kumar, Banerjee, Soumyarup +1 · 1 citation
Mathematics · #11R42 #33E20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT) #Primary 11M06 #Secondary 33C10 #math.CA #math.NT #msc:11M06 #msc:11R42 #msc:33C10 #msc:33E20

paper · pdf · doi:10.48550/arxiv.2105.04141

25 pages

arxiv created 2021/05/10 · arxiv updated 2021/05/11

Abstract

In this article, we study special values of the Dedekind zeta function over an imaginary quadratic field. The values of the Dedekind zeta function at any even integer over any totally real number field is quite well known in literature. In fact, in one of the famous article, Zagier obtained an explicit formula for Dedekind zeta function at point 2 and conjectured an identity at any even values over any number field. We here exhibit the identities for both even and odd values of the Dedekind zeta function over an imaginary quadratic field which are analogous to Ramanujan's identities for even and odd zeta values over \Q. Moreover, any complex zeta values over imaginary quadratic field may also be evaluated from our identities.

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