Czechoslovak Mathematical Journal

, Volume 69, Issue 1, pp 25–37 | Cite as

The Size of the Lerch Zeta-Function at Places Symmetric with Respect to the Line ℜ(s) = 1/2

  • Ramūnas GarunkštisEmail author
  • Andrius Grigutis


Let ζ(s) be the Riemann zeta-function. If t ⩾ 6.8 and σ > 1/2, then it is known that the inequality |ζ(1 − s)| > |ζ(s)| is valid except at the zeros of ζ(s). Here we investigate the Lerch zeta-function L(λ, α, s) which usually has many zeros off the critical line and it is expected that these zeros are asymmetrically distributed with respect to the critical line. However, for equal parameters λ = α it is still possible to obtain a certain version of the inequality |L(λ, λ, \(1 - \bar s\))| > |L(λ, λ, s)|.


Lerch zeta-function functional equation zero distribution 

MSC 2010



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  1. [1]
    H. Alzer: Monotonicity properties of the Riemann zeta function. Mediterr. J. Math. 9 (2012), 439–452.MathSciNetzbMATHGoogle Scholar
  2. [2]
    T. M. Apostol: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer, New York, 1976.Google Scholar
  3. [3]
    B. C. Berndt: On the zeros of a class of Dirichlet series I. Ill. J. Math. 14 (1970), 244–258.MathSciNetzbMATHGoogle Scholar
  4. [4]
    R. D. Dixon, L. Schoenfeld: The size of the Riemann zeta-function at places symmetric with respect to the point 1 2.. Duke Math. J. 33 (1966), 291–292.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    R. Garunkštis: On a positivity property of the Riemann -function. Lith. Math. J. 42 (2002), 140–145; and Liet. Mat. Rink. 42 (2002), 179–184.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    R. Garunkštis, A. Grigutis: The size of the Selberg zeta-function at places symmetric with respect to the line Re(s) = 1/2. Result. Math. 70 (2016), 271–281.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    R. Garunkštis, A. Laurinčikas: On zeros of the Lerch zeta-function. Number Theory and Its Applications. Proc. of the Conf. Held at the RIMS, Kyoto, 1997 (S. Kanemitsu et al., eds.). Dev. Math. 2, Kluwer Academic Publishers, Dordrecht, 1999, pp. 129–143.Google Scholar
  8. [8]
    R. Garunkštis, A. Laurinčikas, J. Steuding: On the mean square of Lerch zeta-functions. Arch. Math. 80 (2003), 47–60.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    R. Garunkštis, J. Steuding: On the zero distributions of Lerch zeta-functions. Analysis, München 22 (2002), 1–12.MathSciNetzbMATHGoogle Scholar
  10. [10]
    R. Garunkštis, J. Steuding: Do Lerch zeta-functions satisfy the Lindelöf hypothesis? Analytic and Probabilistic Methods in Number Theory. Proc. of the Third International Conf. in Honour of J. Kubilius, Palanga, 2001 (A. Dubickas, et al., eds.). TEV, Vilnius, 2002, pp. 61–74.zbMATHGoogle Scholar
  11. [11]
    R. Garunkštis, R. Tamošiū nas: Symmetry of zeros of Lerch zeta-function for equal parameters. Lith. Math. J. 57 (2017), 433–440.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A. Grigutis, D. Šiaučiūnas: On the modulus of the Selberg zeta-functions in the critical strip. Math. Model. Anal. 20 (2015), 852–865.MathSciNetCrossRefGoogle Scholar
  13. [13]
    A. Hinkkanen: On functions of bounded type. Complex Variables, Theory Appl. 34 (1997), 119–139.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J. Lagarias: On a positivity property of the Riemann -function. Acta Arith. 89 (1999), 217–234; correction ibid. 116 (2005), 293–294.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    A. Laurinčikas, R. Garunkštis: The Lerch Zeta-Function. Kluwer Academic Publishers, Dordrecht, 2002.zbMATHGoogle Scholar
  16. [16]
    Y. Matiyasevich, F. Saidak, P. Zvengrowski: Horizontal monotonicity of the modulus of the zeta function, L-functions, and related functions. Acta Arith. 166 (2014), 189–200.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    S. Nazardonyavi, S. Yakubovich: Another proof of Spira’s inequality and its application to the Riemann hypothesis. J. Math. Inequal. 7 (2013), 167–174.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    F. Saidak, P. Zvengrowski: On the modulus of the Riemann zeta function in the critical strip. Math. Slovaca 53 (2003), 145–172.MathSciNetzbMATHGoogle Scholar
  19. [19]
    J. Sondow, C. Dumitrescu: A monotonicity property of Riemann’s xi function and a reformulation of the Riemann hypothesis. Period. Math. Hung. 60 (2010), 37–40.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    R. Spira: An inequality for the Riemann zeta function. DukeMath. J. 32 (1965), 247–250.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    R. Spira: Calculation of the Ramanujan Dirichleseries. Math. Comput. 27 (1973), 379–385.zbMATHGoogle Scholar
  22. [22]
    R. Spira: Zeros of Hurwitz zeta functions. Math. Comput. 30 (1976), 863–866.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    E. C. Titchmarsh: The Theory of Functions. Oxford University Press, Oxford, 1939.zbMATHGoogle Scholar
  24. [24]
    E. C. Titchmarsh: The Theory of the Riemann Zeta-Function. Oxford Science Publications, Oxford University Press, New York, 1986.Google Scholar
  25. [25]
    T. S. Trudgian: A short extension of two of Spira’s results. J. Math. Inequal. 9 (2015), 795–798.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsVilnius UniversityVilniusLithuania

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