On universality for linear combinations of L-functions
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In the present paper, we consider the universality property in the Voronin sense for certain combinations of L-functions with general Dirichlet series as coefficients. In addition, we present some interesting examples of zeta and L-functions which can be expressed in this form. More precisely, we obtain the universality theorem for zeta functions associated to certain arithmetic functions, zeta functions associated to symmetric matrices and Euler–Zagier double zeta and L-functions.
KeywordsHybrid universality Linear combination of L-functions Zeros of L-functions Zeta functions associated to arithmetic functions Zeta functions associated to symmetric matrices Euler–Zagier double zeta-function
Mathematics Subject Classification (2000)11M32 11M35 11M41
We would like to thank the anonymous referee for very useful comments and remarks.
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- 1.Akiyama, S., Ishikawa, H.: On analytic continuation of multiple L-functions and related zeta-functions. Analytic number theory (Beijing/Kyoto, 1999), Dev. Math., 6, pp. 1–16, Kluwer Acad. Publ., Dordrecht (2002)Google Scholar
- 2.Bagchi, B.: The statistical behaviour and universality properties of the Riemann Zeta-function and other allied dirichlet series. Ph.D. thesis, Calcutta, Indian Statistical Institute (1981)Google Scholar
- 4.Gonek, S.M.: Analytic properties of Zeta and L-functions. Ph.D. thesis, Universality of Michigan (1979)Google Scholar
- 10.Laurinčikas A.: The universality of Dirichlet series attached to finite abelian groups, Number theory (Turku, 1999), pp. 179–211. de Gruyter, Berlin (2001)Google Scholar
- 12.Laurinčikas, A., Matsumoto, K.: Joint value-distribution theorems for the Lerch zeta-functions III. In: “Analytic and probabilistic methods in number theory”, Proceedings of the 4th International Conference in Honour of J. Kubilius, A. Laurinčikas, E. Manstavičius (eds.), pp. 87–98. TEV, Vilnius (2007)Google Scholar
- 14.Ram Murty M., Sinha K.: Multiple Hurwitz zeta functions. Multiple Dirichlet series, automorphic forms, and analytic number theory, Proceedings of Symposia in Pure Mathematics 75, 135–156 (2006)Google Scholar
- 18.Pańkowski Ł.: Hybrid joint universality theorem for L-functions without the Euler product (submitted)Google Scholar
- 23.Titchmarsh, E.C.: In: Heath-Brown D. R. (ed.) The theory of the Riemann zeta-function. 2nd edn. The Clarendon Press, Oxford University Press, New York (1986)Google Scholar