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The joint universality for pairs of zeta functions in the Selberg class

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Abstract

We establish a joint universality theorem for pairs of functions in the Selberg class under certain conditions. This theorem generalizes and unifies several previous results, which were shown individually. We also give further examples of pairs of jointly universal L-functions, and actually extend the known universality theorem for the symmetric power L-function \({L(s, \mathrm{sym}^m f)}\) associated to a holomorphic Hecke eigen cusp form f for \({\mathrm{SL}_{2} (\mathbb{Z})}\) with \({1 \le m \le 4}\).

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References

  1. B. Bagchi, The statistical behavior and universality properties of the Riemann zeta-function and other allied Dirichlet series, Thesis, Indian Statistical Institute (Calcutta, 1981).

  2. Bagchi B.: A joint universality theorem for Dirichlet L-functions. Math. Z. 181, 319–334 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauer H.: The value distribution of Artin L-series and zeros of zeta-functions. J. Number Theory 98, 254–279 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bohr H., Courant R.: Neue Anwendungen der Theorie der Diophantischen auf die Riemannsche Zetafunktion. J. Reine Angew. Math. 144, 249–274 (1914)

    MathSciNet  MATH  Google Scholar 

  5. Bombieri E., Hejhal D. A.: On the distribution of zeros of linear combinations of Euler products. Duke Math. J. 80, 821–862 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cogdell J., Michel P.: On the complex moments of symmetric power L-functions at s = 1. Int. Math. Res. Not. 31, 1561–1617 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Duke W., Kowalski E.: A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, with an appendix by Dinakar Ramakrishnan. Invent. Math. 139, 1–39 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. S. M. Gonek, Analytic properties of zeta and L-functions, Thesis, University of Michigan (1979).

  9. J. Kaczorowski, Axiomatic theory of L-functions: the Selberg class, in: Analytic Number Theory, Lecture Notes in Math. 1891, Springer (Berlin, 2006), pp. 133–209.

  10. A. Laurinčikas, Limit Theorems for the Riemann Zeta-Function, Mathematics and its Applications, 352, Kluwer Academic Publishers Group (Dordrecht, 1996).

  11. Laurinčikas A., Matsumoto K.: The joint universality of twisted automorphic L-functions. J. Math. Soc. Japan 56, 923–939 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Y. Lee, T. Nakamura and L. Pańkowski, Selberg’s orthonormality conjecture and joint universality of L-functions, https://arxiv.org/abs/1503.03620, version 3.

  13. Li H., Wu J.: The universality of symmetric power L-functions and their Rankin–Selberg L-functions. J. Math. Soc. Japan 59, 371–392 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu J., Ye Y.: Perron’s formula and the prime number theorem for automorphic L-functions. Pure Appl. Math. Q. 3, 481–497 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mishou H.: Joint universality theorems for pairs of automorphic zeta functions. Math. Z. 277, 1113–1154 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mishou H., Nagoshi H.: Functional distribution of \({L(s,\chi_{d})}\) with real characters and denseness of quadratic class numbers. Trans. Amer. Math. Soc. 358, 4343–4366 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ram Murty M., Zaharescu A.: Explicit formulas for the pair correlation of zeros of functions in the Selberg class. Forum Math. 14, 65–83 (2002)

    MathSciNet  MATH  Google Scholar 

  18. Nagoshi H.: On the universality for L-functions attached to Maass forms. Analysis 25, 1–22 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Nagoshi, Value-distribution of Rankin–Selberg L-functions, in: New Directions in Value-Distribution Theory of Zeta and L-Functions, Shaker Verlag (Aachen, 2009), pp. 275–287.

  20. Nagoshi H.: Joint value-distribution of L-functions and discrepancy of Hecke eigenvalues. Lithuanian Math. J. 56, 325–356 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nagoshi H., Steuding J.: Universality for L-functions in the Selberg class. Lithuanian Math. J. 50, 293–311 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Perelli A.: A survey of the Selberg class of L-functions. I. Milan J. Math. 73, 19–52 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Selberg, Old and new conjectures and results about a class of Dirichlet series, in: Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Univ. Salerno (Salerno, 1992), pp. 367–385.

  24. J. Steuding, Value-Distribution of L-Functions, Lecture Notes in Math. 1877, Springer (Berlin, 2007).

  25. S. M. Voronin, Theorem on the universality of the Riemann zeta function, Izv. Acad. Nauk. SSSR Ser. Mat., 39 (1975), 475–486 (in Russian); Math. USSR Izv., 9 (1975), 443–453.

  26. Voronin S. M.: Analytic properties of Dirichlet generating functions of arithmetic objects. Math. Notes 24, 966–969 (1978)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Information Environment, Tokyo Denki University, 2-1200 Muzai Gakuendai, Inzai-city, Chiba, 270-1382, Japan

    H. Mishou

  2. Faculty of Science and Technology, Gunma University, Kiryu, Gunma, 376-8515, Japan

    H. Nagoshi

Authors
  1. H. Mishou
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  2. H. Nagoshi
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Correspondence to H. Mishou.

Additional information

This work was supported in part by JSPS KAKENHI Grant Numbers 25800031 and 25400005.

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Mishou, H., Nagoshi, H. The joint universality for pairs of zeta functions in the Selberg class. Acta Math. Hungar. 151, 282–327 (2017). https://doi.org/10.1007/s10474-017-0696-4

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  • DOI: https://doi.org/10.1007/s10474-017-0696-4

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