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The Zeros of the Lerch Zeta-Function are Uniformly Distributed Modulo One

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Ukrainian Mathematical Journal Aims and scope

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

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References

  1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Wiley-Interscience Publ., New York (1972).

    MATH  Google Scholar 

  2. A. Akbary and M. R. Murty, “Uniform distribution of zeros of Dirichlet series,” Anatomy of Integers, CRM Proc. Lecture Notes, 46, Amer. Math. Soc., Providence, RI (2008), pp. 143–158.

  3. P. D. T. A. Elliott, “The Riemann zeta function and coin tossing,” J. Reine Angew. Math., 254, 100–109 (1972).

    MathSciNet  MATH  Google Scholar 

  4. K. Ford, K. Soundararajan, and A. Zaharescu, “On the distribution of imaginary parts of zeros of the Riemann zeta function. II,” Math. Ann., 343, No. 3, 487–505 (2009).

    Article  MathSciNet  Google Scholar 

  5. A. Fujii, “On the uniformity of the distribution of zeros of the Riemann zeta function,” J. Reine Angew. Math., 302, 167–205 (1978).

    MathSciNet  MATH  Google Scholar 

  6. R. Garunkštis, “The universality theorem with weight for the Lerch zeta-function,” in: New Trends in Probability and Statistics, vol. 4 (Palanga, 1996), VSP, Utrecht (1997), pp. 59–67.

  7. R. Garunkštis and A. Laurinčikas, “On zeros of the Lerch zeta-function,” in: Number Theory and Its Applications, S. Kanemitsu and K. Gyory (eds.), Kluwer Academic Publishers, Dordrecht (1999), pp. 129–143.

    Google Scholar 

  8. R. Garunkštis and A. Laurinčikas, “The Lerch zeta-function,” Integral Transform. Spec. Funct., 10, 211–226 (2000).

    Article  MathSciNet  Google Scholar 

  9. R. Garunkštis and J. Steuding, “On the zero distributions of Lerch zeta-functions,” Analysis (Munich), 22, 1–12 (2002).

    MathSciNet  MATH  Google Scholar 

  10. R. Garunkštis, A. Laurinčikas, and J. Steuding, “On the mean square of Lerch zeta-functions,” Arch. Math. (Basel), 80, 47–60 (2003).

    Article  MathSciNet  Google Scholar 

  11. R. Garunkštis, J. Steuding, and R. Šimėnas, “The a-points of the Selberg zeta-function are uniformly distributed modulo one,” Illinois J. Math., 58, No. 1, 207–218 (2014).

    Article  MathSciNet  Google Scholar 

  12. R. Garunkštis and J. Steuding, “Do Lerch zeta-functions satisfy the Lindelöf hypothesis?,” Analytic and Probabilistic Methods in Number Theory (Palanga, 2001), TEV, Vilnius (2002), pp. 61–74.

  13. R. Garunkštis and R. Tamošiūnas, “Symmetry of zeros of Lerch zeta-function for equal parameters,” Lith. Math. J., 57, 433–440 (2017).

    Article  MathSciNet  Google Scholar 

  14. E. Hlawka, “Über die Gleichverteilung gewisser Folgen, welche mit den Nullstellen der Zetafunktionen zusammenhängen,” Österreich. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II, 184, 459–471 (1975).

    MathSciNet  MATH  Google Scholar 

  15. A. Laurinčikas, “The universality of the Lerch zeta-function,” Lith. Math. J., 37, 275–280 (1997).

    Article  MathSciNet  Google Scholar 

  16. A. Laurinčikas and R. Garunkštis, The Lerch Zeta-Function, Kluwer Academic Publishers, Dordrecht (2002).

    MATH  Google Scholar 

  17. M. Lerch, “Note sur la fonction \( K\left(z,x,s\right)={\sum}_{k=0}^{\infty }{e}^{2 k\pi ix}{\left(z+k\right)}^{-s} \),” Acta Math., 11, 19–24 (1887).

  18. Y. Lee, T. Nakamura, and Ł. Pańkowski, “Joint universality for Lerch zeta-functions,” J. Math. Soc. Japan, 69, 153–161 (2017).

  19. N. Levinson, “Almost all roots of 𝜁(s) = a are arbitrarily close to σ = 1/2,” Proc. Natl. Acad. Sci. USA, 72, 1322–1324 (1975).

    Article  MathSciNet  Google Scholar 

  20. 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, MIT Press, Cambridge, Mass.-London (1974), pp. 434–458.

  21. J. Steuding, “The roots of the equation 𝜁(s) = a are uniformly distributed modulo one,” Analytic and Probabilistic Methods in Number Theory, TEV, Vilnius (2012), pp. 243–249.

    MATH  Google Scholar 

  22. R. Spira, “Zeros of Hurwitz zeta-functions,” Math. Comp., 136, No. 136, 863–866 (1976).

    Article  MathSciNet  Google Scholar 

  23. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function. 2nd edition, Ed. D. R. Heath-Brown, The Clarendon Press, Oxford University Press, New York (1986).

  24. H. Weyl, “Sur une application de la théorie des nombres à la mécaniques statistique et la théorie des pertubations,” Enseign. Math., 16, 455–467 (1914).

    MATH  Google Scholar 

  25. H. Weyl, “Über die Gleichverteilung von Zahlen mod. Eins,” Math. Ann., 77, 313–352 (1916).

    Article  MathSciNet  Google Scholar 

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Correspondence to T. Panavas.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 9, pp. 1170–1180, September, 2021. Ukrainian DOI: 10.37863/umzh.v73i9.893.

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Garunkštis, R., Panavas, T. The Zeros of the Lerch Zeta-Function are Uniformly Distributed Modulo One. Ukr Math J 73, 1359–1370 (2022). https://doi.org/10.1007/s11253-022-01999-2

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  • DOI: https://doi.org/10.1007/s11253-022-01999-2

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