Skip to main content
Log in

On complex zeros off the critical line for non-monomial polynomial of zeta-functions

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this paper, we show that any polynomial of zeta or L-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler–Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindelöf hypothesis for the Riemann zeta-function is equivalent to the Lindelöf hypothesis for zeta-functions mentioned above despite of the existence of the zeros off the critical line. Next we prove that the Barnes multiple zeta-functions associated to rational or transcendental parameters have infinitely many zeros off the critical line. By using this fact, we show that the Shintani multiple zeta-functions have infinitely many complex zeros under some conditions. As corollaries, we show that the Mordell multiple zeta-functions, the Euler–Zagier–Hurwitz type of multiple zeta-functions and the Witten multiple zeta-functions have infinitely many complex zeros off the critical line.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akiyama, S., Egami, S., Tanigawa, Y.: Analytic continuation of multiple zeta functions and their values at non-positive integers. Acta Arith. 98(2), 107–116 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Akiyama, S., Ishikawa, H.: On analytic continuation of multiple L-functions and related zeta-functions. In: Jia, C., Matsumoto, K. (eds.) Analytic number theory. Development in mathematics, vol. 6, pp 1–16. Springer, Dordrecht (2002)

  3. Akiyama, S., Tanigawa, Y.: Multiple zeta values at non-positive integers. Ramanujan J 5(4), 327–351 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Atkinson, F.V.: The mean-value of the Riemann zeta function. Acta Math. 81, 353–376 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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)

  6. Barnes, E.W.: On the theory of multiple gamma function. Trans. Camb. Philos. Soc. 19, 374–425 (1904)

    Google Scholar 

  7. Bohr, H.: Über das Verhalten von \(\zeta (s)\) in der Halbebene \(\sigma >1\). Nachr. Akad. Wiss. Göttingen II Math.-phys. Kl. 409–428 (1911)

  8. Bohr, H.: Über eine quasi-periodische eigenschaft Dirichletscher reihen mit anwendung auf die Diriehletschen L-Funktionen. Math. Ann. 85(1), 115–122 (1922)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cassou-Noguès, P.: Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques. Invent. Math. 51, 29–59 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gonek, S.M.: Analytic Properties of Zeta and L-functions. Ph.D. Thesis, Universality of Michigan (1979)

  11. Gunnells, E., Sczech, R.: Evaluation of Dedekind sums, Eisenstein cocycles, and special values of \(L\)-functions. Duke Math. J. 118(2), 229–260 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hejhal, D.: On a result of G. Polya concerning the Riemann \(\xi \)-function. J. Anal. Math. 55, 59–95 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hida, H.: Elementary Theory of L-functions and Eisenstein Series. London Math. Soc. Student Texts 26, Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  14. Huxley, M.N.: Exponential sums and the Riemann zeta function. V. Proc. London Math. Soc. (3) 90(1), 1–41 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hoffman, M.E.: Multiple harmonic series. Pac. J. Math. 152(2), 275–290 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ibukiyama, T., Saito, H.: On zeta functions associated to symmetric matrices, I: An explicit form of zeta functions. Am. J. Math. 117(5), 1097–1155 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Imai, H.: On the construction of p-adic L-functions. Hokkaido Math. J. 10, 249–253 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ishikawa, H., Matsumoto, K.: On the estimation of the order of Euler–Zagier multiple zeta-functions. Illinois J. Math. 47(4), 1151–1166 (2003)

    MathSciNet  MATH  Google Scholar 

  19. Kačinskaitė, R., Steuding, J., Šiaučiūnas, D., Šleževičienė, R.: On polynomials in Dirichlet series. Fizikos ir Matematikos Fakulteto Mokslinio Seminaro Darbai 7, 26–32 (2004)

    MathSciNet  MATH  Google Scholar 

  20. Kaczorowski, J., Kulas, M.: On the non-trivial zeros off line for L-functions from extended Selberg class. Monatshefte Math. 150(3), 217–232 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kelliher, J.P., Masri, R.: Analytic continuation of multiple Hurwitz zeta functions. Math. Proc. Camb. Philos. Soc. 145(3), 605–617 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ki, H.: Zeros of zeta functions. Comment. Math. Univ. St. Pauli 60(1–2), 99–117 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Kiuchi, I., Tanigawa, Y.: Bounds for triple zeta-functions. Indag. Math. (N.S.) 19(1), 97–114 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Komori, Y., Matsumoto, K., Tsumura, H.: On Witten multiple zeta-functions associated with semisimple Lie algebras II. J. Math. Soc. Jpn. 62(2), 355–394 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Laurinčikas, A., Garunkšitis, R.: The Lerch Zeta-Function. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

  26. Lagarias, J.C., Suzuki, M.: The Riemann hypothesis for certain integrals of Eisenstein series. J. Number Theory 118(1), 98–122 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Matsumoto, K.: The analytic continuation and the asymptotic behavior of certain multiple zeta-functions. I. J. Number Theory 101(2), 223–243 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Matsumoto, K.: Functional equations for double zeta-functions. Math. Proc. Camb. Philos. Soc. 136(1), 1–7 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Matsumoto, K.: Analytic theory of multiple zeta-functions and its applications. Sugaku Expos. 23(2), 143–167 (2010)

    MATH  Google Scholar 

  30. Mordell, L.J.: On the evaluation of some multiple series. J. Lond. Math. Soc. 33, 368–371 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  31. Murty M.R., Sinha, K.: Multiple Hurwitz zeta functions, multiple Dirichlet series, automorphic forms, and analytic number theory. In: Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society, Providence, RI, vol. 75, pp. 135–156 (2006)

  32. Nakamura, T.: Zeros and the universality for the Euler–Zagier–Hurwitz type of multiple zeta-functions. Bull. Lond. Math. Soc. 41(4), 691–700 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Nakamura, T., Pańkowski, Ł.: On universality for linear combinations of L-functions. Mon. Math. 165(3), 433–446 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nakamura, T., Pańkowski, Ł.: Applications of hybrid universality to multivariable zeta-functions. J. Number Theory 131(11), 2151–2161 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Osgood, B., Phillips, R., Sarnak, P.: Extremals of determinants of Laplacians. J. Funct. Anal. 80(1), 148–211 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  36. Pańkowski, Ł.: Hybrid joint universality theorem for the Dirichlet L-functions. Acta Arith. 141(1), 59–72 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pańkowski, Ł.: Hybrid joint universality theorem for L-functions without the Euler product. Integral Transforms Spec. Funct. 24(1), 39–49 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Steuding, J.: Value-distribution of L-functions. In: Lecture Notes in Mathematics, vol. 1877. Springer, Berlin (2007)

  39. Shintani, T.: On evaluation of zeta functions of totally real algebraic number fields at nonpositive integers. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23, 393–417 (1976)

    MathSciNet  MATH  Google Scholar 

  40. Srivastava, H.M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer, Berlin (2001)

    Book  MATH  Google Scholar 

  41. Taylor, P.R.: On the Riemann zeta function. Quart. J. Oxford 19, 1–21 (1945)

    Article  MATH  Google Scholar 

  42. Taylor, M.E.: Partial Differential Equations II: Qualitative Studies of Linear Equations, Second Edition. Applied Mathematical Sciences, vol. 116. Springer, New York (2011)

    Google Scholar 

  43. Titchmarsh, E.C.: The Theory of Functions. Oxford University Press, Oxford (1939)

    MATH  Google Scholar 

  44. Titchmarsh, E.C.: The Theory of the Riemann Zeta-function, Second edition. Edited and with a preface by D.R. Heath-Brown. The Clarendon Press, Oxford University Press, New York (1986)

  45. Tornheim, L.: Harmonic double series. Am. J. Math. 72, 303–314 (1950)

    Article  MathSciNet  MATH  Google Scholar 

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

  47. Witten, E.: On quantum gauge theories in two dimensions. Commun. Math. Phys. 141(1), 153–209 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  48. Zagier, D.: Values of zeta functions and their applications, First European Congress of Math., Paris, vol. II, Progress in Math, Birk ä user, vol. 120, pp. 497–512 (1994)

  49. Zhao, J.: Analytic continuation of multiple zeta functions. Proc. Am. Math. Soc. 128(5), 1275–1283 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Takashi Nakamura.

Additional information

The first author was partially supported by JSPS Grants 21740024. The second author was partially supported by the Grant No. N N201 6059 40 from National Science Centre.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nakamura, T., Pańkowski, Ł. On complex zeros off the critical line for non-monomial polynomial of zeta-functions. Math. Z. 284, 23–39 (2016). https://doi.org/10.1007/s00209-016-1643-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-016-1643-8

Keywords

Mathematics Subject Classification

Navigation