Illinois Journal of Mathematics

The $a$-points of the Selberg zeta-function are uniformly distributed modulo one

Ramūnas Garunkštis, Jörn Steuding, and Raivydas Šimėnas

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Abstract

Let $Z(s)$ be the Selberg zeta-function associated with a compact Riemann surface. We prove that the imaginary parts of the nontrivial $a$-points of $Z(s)$ are uniformly distributed modulo one. We also consider the question whether the eigenvalues of the corresponding Laplacian are uniformly distributed modulo one.

Article information

Source
Illinois J. Math., Volume 58, Number 1 (2014), 207-218.

Dates
First available in Project Euclid: 1 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1427897174

Digital Object Identifier
doi:10.1215/ijm/1427897174

Mathematical Reviews number (MathSciNet)
MR3331847

Zentralblatt MATH identifier
1319.11061

Subjects
Primary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas

Citation

Garunkštis, Ramūnas; Steuding, Jörn; Šimėnas, Raivydas. The $a$-points of the Selberg zeta-function are uniformly distributed modulo one. Illinois J. Math. 58 (2014), no. 1, 207--218. doi:10.1215/ijm/1427897174. https://projecteuclid.org/euclid.ijm/1427897174


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References

  • A. Akbary and M. R. Murty, Uniform distribution of zeros of Dirichlet series, Anatomy of integers (J.-M. De Koninck \betal, eds.), CRM workshop, Montreal, Canada, March 13–17, 2006, CRM Proceedings and Lecture Notes, vol. 46, Amer. Math. Soc., Providence, RI, 2008, pp. 143–158.
  • P. Drungilas, R. Garunkštis and A. Kač\.enas, Universality of the Selberg zeta-function for the modular group, Forum Math. 25 (2013), no. 3, 533–564.
  • P. D. T. A. Elliott, The Riemann zeta function and coin tossing, J. Reine Angew. Math. 254 (1972), 100–109.
  • K. Ford, K. Soundararajan and A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function. II, Math. Ann. 343 (2009), no. 3, 487–505.
  • A. Fujii, On the uniformity of the distribution of the zeros of the Riemann zeta-function, J. Reine Angew. Math. 302 (1978), 167–205.
  • R. Garunkštis, On the Backlund equivalent for the Lindelöf hypothesis, Adv. Stud. Pure Math. 49 (2007), 91–104.
  • R. Garunkštis and R. Šim\.enas, The $a$-values of the Selberg zeta-function, Lith. Math. J. 52 (2012), no. 2, 145–154.
  • G. H. Hardy and M. Riesz, The general theory of Dirichlet's series, Cambridge University Press, Cambridge, MA, 1915.
  • D. A. Hejhal, The Selberg trace formula for $\operatorname{PSL}(2, \mathbb{R})$, vol. 1, Lecture Notes in Mathematics, vol. 548, Springer, Berlin, 1976.
  • E. Hlawka, Über die Gleichverteilung gewisser Folgen, welche mit den Nullstellen der Zetafuncktionen zusammenhängen, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 184 (1975), 459–471.
  • E. Landau, Über den Wertevorrat von $\zeta(s)$ in der Halbebene $\sigma>1$, Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl. (1933), 81–91.
  • H. Rademacher, Fourier analysis in number theory, Cornell Univ., Ithica, NY, 1956, 25 pages. Also in: Collected papers of Hans Rademacher, vol. II, Mathematicians of Our Time, vol. 4, MIT Press, Cambridge, MA, 1974, pp. 434–458.
  • B. Randol, Small eigenvalues of the Laplace operator on compact Riemann surfaces, Bull. Amer. Math. Soc. (N.S.) 80 (1974), 996–1000.
  • B. Randol, On the asymptotic distribution of closed geodesics on compact Riemann surfaces, Trans. Amer. Math. Soc. 233 (1977), 241–247.
  • B. Randol, The Riemann hypothesis for Selberg's zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator, Trans. Amer. Math. Soc. 236 (1978), 209–223.
  • J. Steuding, The roots of the equation $\zeta(s)=a$ are uniformly distributed modulo one, Anal. probab. methods number theory (E. Manstavičius \betal, eds.), TEV, Vilnius, 2012, pp. 243–249.
  • E. C. Titchmarsh, The theory of the Riemann zeta-function, Clarendon Press, Oxford, 1988.
  • H. Weyl, Sur une application de la théorie des nombres à la mécaniques statistique et la théorie des pertubations, Enseign. Math. (2) 16 (1914), 455–467.
  • H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313–352.