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Mathematische Annalen

, Volume 371, Issue 3–4, pp 1191–1227 | Cite as

On the location of the zero-free half-plane of a random Epstein zeta function

  • Andreas Strömbergsson
  • Anders SödergrenEmail author
Article

Abstract

In this note we study, for a random lattice L of large dimension n, the supremum of the real parts of the zeros of the Epstein zeta function \(E_n(L,s)\) and prove that this random variable scaled by \(n^{-1}\) has a limit distribution, which we give explicitly. This limit distribution is studied in some detail; in particular we give an explicit formula for its distribution function. Furthermore, we obtain a limit distribution for the frequency of zeros of \(E_n(L,s)\) in vertical strips contained in the half-plane \(\mathfrak {R}s>\frac{n}{2}\).

Notes

Acknowledgements

We are grateful to Daniel Fiorilli and Svante Janson for inspiring discussions and helpful remarks. We are also grateful to the referee for asking about the density of zeros; this inspired us to add Theorem 2 to the paper. The second author thanks the Institute for Advanced Study for providing excellent working conditions.

Supplementary material

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References

  1. 1.
    Bateman, P.T., Grosswald, E.: On Epstein’s zeta function. Acta Arith. 9, 365–373 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bernays, P.: Über die Darstellung von positiven, ganzen Zahlen durch die primitiven, binären quadratischen Formen einer nicht-quadratischen Diskriminante, Dissertation, Göttingen (1912)Google Scholar
  3. 3.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics. Wiley, New York (1999)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bohr, H., Jessen, B.: Über die Werteverteilung der Riemannschen Zetafunktion. Acta Math. 54(1), 1–35 (1930)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bombieri, E., Ghosh, A.: Around the Davenport-Heilbronn function, Uspekhi Mat. Nauk 66(2), 15–66 (2011); translation in Russian Math. Surveys 66(2), 221–270 (2011)Google Scholar
  6. 6.
    Bombieri, E., Hejhal, D.A.: On the distribution of zeros of linear combinations of Euler products. Duke Math. J. 80(3), 821–862 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bombieri, E., Mueller, J.: On the zeros of certain Epstein zeta functions. Forum Math. 20(2), 359–385 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Davenport, H., Heilbronn, H.: On the zeros of certain Dirichlet series. J. Lond. Math. Soc. 11, 181–185, 307–312 (1936)Google Scholar
  9. 9.
    Edelman, A., Kostlan, E.: How many zeros of a random polynomial are real? Bull. Amer. Math. Soc. (N.S.) 32(1), 1–37 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Feller, W.: An introduction to probability theory and its applications. vol. II, second edn. Wiley, New York (1971)zbMATHGoogle Scholar
  11. 11.
    Gonek, S., Lee, Y.: Zero-density estimates for Epstein zeta functions. Q. J. Math. 68(2), 301–344 (2017)Google Scholar
  12. 12.
    Hejhal, D.A.: On a result of Selberg concerning zeros of linear combinations of \(L\)-functions. Int. Math. Res. Not. 2000(11), 551–577 (2000)Google Scholar
  13. 13.
    Jessen, B.: Über die Nullstellen einer analytischen fastperiodischen Funktion. Eine Verallgemeinerung der Jensenschen Formel. Math. Ann. 108(1), 485–516 (1933)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jessen, B.: The theory of integration in a space of an infinite number of dimensions. Acta Math. 63(1), 249–323 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jutila, M., Srinivas, K.: Gaps between the zeros of Epstein’s zeta-functions on the critical line. Bull. Lond. Math. Soc. 37(1), 45–53 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kallenberg, O.: Foundations of Modern Probability: Probability and its Applications, 2nd edn. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  17. 17.
    Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications 54. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  18. 18.
    Lee, Y.: On the zeros of Epstein zeta functions. Forum Math. 26(6), 1807–1836 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mukhopadhyay, A., Rajkumar, K., Srinivas, K.: On the zeros of the Epstein zeta function. In: Number Theory. Ramanujan Math. Soc. Lect. Notes Ser., vol. 15. Ramanujan Math. Soc., Mysore, pp. 73–87 (2011)Google Scholar
  20. 20.
    Pall, G.: The distribution of integers represented by binary quadratic forms. Bull. Amer. Math. Soc. 49, 447–449 (1943)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)zbMATHGoogle Scholar
  22. 22.
    Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman and Hall, New York (1994)zbMATHGoogle Scholar
  23. 23.
    Sarnak, P., Strömbergsson, A.: Minima of Epstein’s zeta function and heights of flat tori. Invent. Math. 165(1), 115–151 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Siegel, C.L.: A mean value theorem in geometry of numbers. Ann. Math. 46, 340–347 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Stark, H.M.: On the zeros of Epstein’s zeta function. Mathematika 14, 47–55 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Steuding, J.: On the zero-distribution of Epstein zeta-functions. Math. Ann. 333(3), 689–697 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Södergren, A.: On the Poisson distribution of lengths of lattice vectors in a random lattice. Math. Z. 269(3–4), 945–954 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Södergren, A.: On the value distribution and moments of the Epstein zeta function to the right of the critical strip. J. Number Theory 131(7), 1176–1208 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Södergren, A.: On the distribution of angles between the \(N\) shortest vectors in a random lattice. J. Lond. Math. Soc. (2) 84(3), 749–764 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Terras, A.: Real zeroes of Epstein’s zeta function for ternary positive definite quadratic forms. Ill. J. Math. 23(1), 1–14 (1979)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Terras, A.: Integral formulas and integral tests for series of positive matrices. Pac. J. Math. 89(2), 471–490 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Terras, A.: The minima of quadratic forms and the behavior of Epstein and Dedekind zeta functions. J. Number Theory 12(2), 258–272 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Terras, A.: Harmonic analysis on symmetric spaces and applications II. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  34. 34.
    Wintner, A.: Upon a statistical method in the theory of diophantine approximations. Amer. J. Math. 55(1–4), 309–331 (1933)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  3. 3.Department of Mathematical SciencesChalmers University of Technology and the University of GothenburgGothenburgSweden

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