The zeros of the Lerch zeta-function are uniformly distributed modulo one

  • R. Garunkštis Inst. Math., Vilnius Univ., Lithuania)
  • T. Panavas Inst. Math., Vilnius Univ., Lithuania
Keywords: Lerch zeta-function, zero distribution, uniform distribution

Abstract

UDC 511.311

We prove that the ordinates of the nontrivial zeros of the Lerch zeta-function are uniformly distributed modulo one.

References

M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards, Wiley-Intersci. Publ., New York (1972)

A. Akbary, M. R. Murty, Uniform distribution of zeros of Dirichlet series, Anatomy of Integer, CRM Proc. Lecture Notes, 46, 143 – 158 (2008), https://doi.org/10.1090/crmp/046/10 DOI: https://doi.org/10.1090/crmp/046/10

P. D. T. A. Elliott, The Riemann zeta function and coin tossing, J. reine und angew. Math., 254, 100 – 109 (1972), https://doi.org/10.1515/crll.1972.254.10 DOI: https://doi.org/10.1515/crll.1972.254.100

K. Ford, K. Soundararajan, A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function, II, Math. Ann., 343, 487 – 505 (2009), DOI: https://doi.org/10.1007/s00208-008-0280-x

A. Fujii, On the uniformity of the distribution of zeros of the Riemann zeta function, J. reine und angew. Math., 302, 167 – 205 (1978), https://doi.org/10.1515/crll.1978.302.167 DOI: https://doi.org/10.1515/crll.1978.302.167

R. Garunkštis, The universality theorem with weight for the Lerch zeta-function, New Trends in Probability and Statistics, vol. 4 (Palanga, 1996), VSP, Utrecht (1997). DOI: https://doi.org/10.1515/9783110944648.59

R. Garunkštis, A. Laurinčikas, On zeros of the Lerch zeta-function, Number Theory and Its Applications, S. Kanemitsu, K. Gyory (eds.), Kluwer Acad. Publ., 129 – 143 (1999), https://doi.org/10.1080/10652460008819287

R. Garunkštis, A. Laurinčikas, The Lerch zeta-function, Integral Transforms Spec. Funct., 10, 211 – 226 (2000), https://doi.org/10.1080/10652460008819287 DOI: https://doi.org/10.1080/10652460008819287

R. Garunkštis, J. Steuding, On the zero distributions of Lerch zeta-functions, Analysis, 22, 1 – 12 (2002), https://doi.org/10.1524/anly.2002.22.1.1 DOI: https://doi.org/10.1524/anly.2002.22.1.1

R. Garunkštis, A. Laurinčikas, J. Steuding, On the mean square of Lerch zeta-functions, Arch. Math., 80, 47 – 60 (2003), https://doi.org/10.1007/s000130300005 DOI: https://doi.org/10.1007/s000130300005

R. Garunkštis, J. Steuding, R. Šimėnas, The a-points of the Selberg zeta-function are uniformly distributed modulo one, Illinois J. Math., 58, 207 – 218 (2014). DOI: https://doi.org/10.1215/ijm/1427897174

R. Garunkštis, J. Steuding, Do Lerch zeta-functions satisfy the Lindeloff hypothesis?, Anal. and Probab. Methods Number Theory, Proc. Third Intern. Conf. Honour of J. Kubilius (Palanga, Lithuania, 24 – 28 September 2001), TEV, Vilnius (2002), p. 61 – 74.

R. Garunkštis, R. Tamošiūnas, Symmetry of zeros of Lerch zeta-function for equal parameters, Lith. Math. J., 57, 433 – 440 (2017), https://doi.org/10.1007/s10986-017-9373-0 DOI: https://doi.org/10.1007/s10986-017-9373-0

E. Hlawka, Über die Gleichverteilung gewisser Folgen, welche mit den Nullstellen der Zetafunktion zusammenhängen, Österreich. Akad. Wiss., Math.-Natur. Kl. Abt. II, 184, 459 – 471 (1975).

A. Laurinčikas, The universality of the Lerch zeta-function, Lith. Math. J., 37, 275 – 280 (1997), https://doi.org/10.1007/BF02465359 DOI: https://doi.org/10.1007/BF02465359

A. Laurinčikas, R. Garunkštis, The Lerch zeta-function, Kluwer Acad. Publ., Dordrecht (2002). DOI: https://doi.org/10.1007/978-94-017-6401-8

M. Lerch, ${germ K} left( {w,x,s} right) = sumlimits_{k = 0}^infty {frac{{e^{2kpi ix} }}{{left( {w + k} right)^s }}} $. (French) , Acta Math., 11, 19 – 24 (1887), https://doi.org/10.1007/BF02418041 DOI: https://doi.org/10.1007/BF02612318

Y. Lee, T. Nakamura, Ł. Pa´nkowski, Joint universality for Lerch zeta-functions, J. Math. Soc. Japan, 69, 153 – 161 (2017), https://doi.org/10.2969/jmsj/06910153 DOI: https://doi.org/10.2969/jmsj/06910153

N. Levinson, Almost all root of $zeta (s)=a$ are arbitrarily close to $sigma =1/2$, Proc. Nat. Acad. Sci. USA, 72, 1322 – 1324 (1975), https://doi.org/10.1073/pnas.72.4.1322 DOI: https://doi.org/10.1073/pnas.72.4.1322

H. G. Rademacher, Fourier analysis in number theory, Symp. Harmonic Analysis and Related Integral Transforms (Cornell Univ., Ithaca, N.Y., 1956), Collected Papers of Hans Rademacher, vol. II (1974), p. 434 – 458.

J. Steuding, The roots of the equation $zeta(s)=a$ are uniformly distributed modulo one, Anal. and Probab. Methods Number Theory TEV, Vilnius (2012), p. 243 – 249.

R. Spira, Zeros of Hurwitz zeta-functions, Math. Comput., 136, 863 – 866 (1976), https://doi.org/10.2307/2005407 DOI: https://doi.org/10.1090/S0025-5718-1976-0409382-2

E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., rev. by D. R. Heath-Brown, Oxford Sci. Publ., Clarendon Press, Oxford (1986).

H. Weyl, Sur une application de la th´eorie des nombres `a la m´ecaniques statistique et la th´eorie des pertubations, Enseign. Math., 16, 455 – 467 (1914).

H. Weyl, U¨ ber die Gleichverteilung von Zahlen mod. Eins, Math. Ann., 77, 313 – 352 (1916). DOI: https://doi.org/10.1007/BF01475864

Published
16.09.2021
How to Cite
GarunkštisR., and PanavasT. “The Zeros of the Lerch Zeta-Function Are Uniformly Distributed Modulo One”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 9, Sept. 2021, pp. 1170 -80, doi:10.37863/umzh.v73i9.893.
Section
Research articles