Analytic properties of multiple Dirichlet series associated to additive and Dirichlet characters
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In this article, we study analytic properties of the multiple Dirichlet series associated to additive and Dirichlet characters. For the multiple Dirichlet series associated to additive characters, the meromorphic continuation is established via obtaining translation formulas satisfied by them. We then determine the exact set of singularities of such multiple Dirichlet series. While it seems difficult to obtain a similar translation formula for the multiple Dirichlet series associated to Dirichlet characters, we establish an intrinsic connection between them and the multiple Dirichlet series associated to additive characters. We rely on this connection to investigate the analytic characteristics of the multiple Dirichlet series associated to Dirichlet characters.
Mathematics Subject Classification11M32 32Dxx
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- 2.Akiyama, S., Ishikawa, H.: On Analytic Continuation of Multiple \(L\)-Functions and Related Zeta-Functions, Analytic Number Theory (Beijing/Kyoto, 1999) 1–16, Dev. Math. 6, Kluwer Acad. Publ., Dordrecht (2002)Google Scholar
- 4.Bowman, D., Bradley, D. M.: Multiple Polylogarithms: A Brief Survey, q-Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000), 71–92, Contemp. Math. 291, Am. Math. Soc., Providence, RI (2001)Google Scholar
- 5.Frobenius, G.: Über die Bernoullischen Zahlen und die Eulerschen Polynome, pp. 809–847. Sitzungsberichte der Preussischen Akad. der Wiss. (1910)Google Scholar
- 7.Goncharov, A. B.: Multiple polylogarithms and mixed Tate motives. arXiv:math/0103059
- 10.Saha, B.: On the analytic continuation of multiple Dirichlet series and their singularities. Ph.D. thesis, HBNI (2016)Google Scholar
- 12.Waldschmidt, M.: Multiple Polylogarithms: An Introduction, Number Theory and Discrete Mathematics (Chandigarh, 2000), 112, Trends Math. Birkhäuser, Basel (2002)Google Scholar