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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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.

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

First available in Project Euclid: 1 April 2015

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Zentralblatt MATH identifier

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


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.

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  • 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.