Bilateral zeta functions and their applications

2011/06/09 by Genki Shibukawa, Shibukawa, Genki
Mathematics · #11B68 #11F03 #11M35 #42A16 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Inequalities and Applications #Number Theory (math.NT) #math.CA #math.NT #msc:11B68 #msc:11F03 #msc:11M35 #msc:42A16

paper · pdf · doi:10.48550/arxiv.1106.1754

24 pages, no figures

openalex publication_date 2011/06/09 · arxiv created 2013/03/31 · arxiv updated 2013/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta function. Especially, we prove that the bilateral zeta function has a nice Fourier series expansion and the Barnes zeta function can be expressed as a finite sum of bilateral zeta functions. By these properties of the bilateral zeta functions, We obtain simple proofs of some formulas, for example the reflection formula for the multiple gamma function, the inversion formula of the Dedekind eta function, Ramanujan's formula, Fourier expansion of the Barnes zeta function and multiple Iseki's formula.

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