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Asymptotic free probability for arithmetic functions and factorization of Dirichlet series

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Abstract

In this paper, we study a free-probabilistic model on the algebra of arithmetic functions by considering their asymptotic behavior. As an application, we concentrate on arithmetic functions arising from certain representations attached to the general linear group \(GL_n\). We then study conditions under which a Dirichlet series may be factored into a product of automorphic L-functions using asymptotic freeness.

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Acknowledgments

The authors would like to thank Prof. Yangbo Ye for valuable discussions and advice. The third named author was supported in part by a bi-national US-Israel Science Foundation grant 2010117. He also acknowledges helpful discussions with Professor Daniel Alpay.

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Authors and Affiliations

  1. Department of Mathematics, St. Ambrose University, 418 Ambrose Hall, 518 W. Locust St., Davenport, Iowa, 52803, USA

    Ilwoo Cho

  2. Department of Mathematics, St. Ambrose University, 421 Ambrose Hall, 518 W. Locust St., Davenport, IA, 52803, USA

    Timothy Gillespie

  3. 14 McLean Hall, Iowa City, IA, 52242, USA

    Palle E. T. Jorgensen

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  1. Ilwoo Cho
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  2. Timothy Gillespie
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  3. Palle E. T. Jorgensen
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Correspondence to Ilwoo Cho.

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Cho, I., Gillespie, T. & Jorgensen, P.E.T. Asymptotic free probability for arithmetic functions and factorization of Dirichlet series. Anal.Math.Phys. 6, 255–295 (2016). https://doi.org/10.1007/s13324-015-0117-1

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  • DOI: https://doi.org/10.1007/s13324-015-0117-1

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