A joint discrete limit theorem for Epstein and Hurwitz zeta-functions
Abstract
In the paper, we obtain a joint limit theorem on weak convergence for probability measure defined by discrete shifts of the Epstein and Hurwitz zeta-functions. The limit measure is explicitly given. For the proof, some linear independence restriction is required. The proved theorem extends and continues Bohr–Jessen’s classical results on probabilistic characterization of value distribution for the Riemann zeta-function.
Keyword : Epstein zeta-function, Hurwitz zeta-function, limit theorem, Haar probability measure, weak convergence
How to Cite
Gerges, H., Laurinčikas, A., & Macaitienė, R. (2025). A joint discrete limit theorem for Epstein and Hurwitz zeta-functions. Mathematical Modelling and Analysis, 30(2), 186–202. https://doi.org/10.3846/mma.2025.22109
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J. Hadamard. Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques. Bull. Soc. Math. France, 24:199–220, 1896. https://doi.org/10.24033/bsmf.545
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A. Hurwitz. Einige Eigenschaften der Dirichlet’schen Funktionen f(s) = die bei der Bestimmung der Klassenzahlen binärer quadratischer Formen auftreten. Zeitschr. für Math. und Phys., 27:86–101, 1882. https://doi.org/10.1007/978-3-0348-4161-0_3
H. Iwaniec. Topics in Classical Automorphic Forms, Graduate Studies in Mathematics. American Mathematical Society: Providence, RI, Volume 17, 1997. https://doi.org/10.1090/gsm/017
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E. Kowalski. An Introduction to Probabilistic Number Theory. Cambridge Univ. Press, 2021. https://doi.org/10.1017/9781108888226
A. Laurinčikas. Limit Theorems for the Riemann Zeta-Function. Kluwer, Dordrecht, 1996. https://doi.org/10.1007/978-94-017-2091-5
A. Laurinčikas. A discrete universality theorem for the Hurwitz zeta-function. J. Number Th., 143:232–247, 2014. https://doi.org/10.1016/j.jnt.2014.04.013
A. Laurinčikas. On value distribution of certain Beurling zeta-functions. Mathematics, 12(459):1–15, 2024. https://doi.org/10.3390/axioms13040251
A. Laurinčikas and R. Garunkštis. The Lerch Zeta-Function. Kluwer, Dordrecht, 2002. https://doi.org/10.1007/978-94-017-6401-8
A. Laurinčikas and R. Macaitienė. A Bohr-Jessen type theorem for the Epstein zeta-function. Results in Math., 73(4):147–163, 2018. https://doi.org/10.1007/s00025-018-0909-3
A. Laurinčikas and R. Macaitienė. A Bohr-Jessen type theorem for the Epstein zeta-function. II. Results in Math., 75(25):1–16, 2020. https://doi.org/10.1007/s00025-019-1151-3
A. Laurinčikas and D. Šiaučiūnas. Generalized limit theorem for Mellin transform of the Riemann zeta-function. Axioms, 13(251):1–17, 2024. https://doi.org/10.3390/axioms13040251
H. L. Montgomery. Topics in Multiplicative Number Theory. Springer-Verlag, Berlin, Heidelberg, New York, 1971. https://doi.org/10.1016/j.jnt.2021.03.021
T. Nakamura and L . Pańkowski. On zeros and c-values of Epstein zeta-functions. Šiauliai Math. Semin., 8:181–195, 2013.
Yu. V. Nesterenko. Modular functions and transcendence questions. Sb. Math., 187:1319–1348, 1996. https://doi.org/10.1070/SM1996v187n09ABEH000158
B. Riemann. Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie, pp. 671–680, 1859.
J. Steuding. Value-Distribution of L-Functions. Lecture Notes Math., vol. 1877, Springer, Berlin, Heidelberg, 2007. https://doi.org/10.1007/978-3-540-44822-8
A. Balčiūnas, V. Garbaliauskienė, V. Lukšienė, R. Macaitienė and A. Rimkevičienė. Joint discrete approximation of analytic functions by Hurwitz zeta-functions. Math. Model. Anal., 27(1):88–100, 2022. https://doi.org/10.3846/mma.2022.15068
P. Bateman and E. Grosswald. On Epstein’s zeta function. Acta Arith., 9:365– 373, 1964. https://doi.org/0.4064/aa-9-4-365-373
P. Billingsley. Convergence of Probability Measures. Willey, New York, 1968. https://doi.org/10.1002/9780470316962
H. Bohr. Zur Theorie der Riemann’schen Zetafunktion im kritischen Streifen. Acta Math., 40:67–100, 1916. https://doi.org/10.1007/bf02418541
H. Bohr and B. Jessen. Über die Wertverteilung der Riemanschen Zetafunktion, Erste mitteilung. Acta Math., 54:1–35, 1930. https://doi.org/10.1007/BF02547516
H. Bohr and B. Jessen. Über die Wertverteilung der Riemanschen Zetafunktion, Zweite Mitteilung. Acta Math., 58:1–55, 1932. https://doi.org/10.1007/BF02547773
E. Buivydas, A. Laurinčikas, J. Rašytė and R. Macaitienė. Discrete universality theorems for the Hurwitz zeta-function. J. Approx. Th., 183:1–13, 2014. https://doi.org/10.1016/j.jat.2014.03.006
S. Chowla and A. Selberg. On Epstein’s zeta-function. Journal für die Reine und Angewandte Mathematik, 227:86–110, 1967. https://doi.org/10.1515/crll.1967.227.86
H. Cramér and M. R. Leadbetter. Stationary and Related Stochastic Processes. Wiley, New York, 1967. https://doi.org/10.2307/2344096
C.-J. de la Vallée Poussin. Recherches analytiques la thorie des nombres premiers. Ann. Soc. Scient. Bruxelles, 20:183–256, 1896.
E. Elizalde. Multidimensional extension of the generalized Chowla–Selberg formula. Comm. Math. Phys., 198:83–95, 1998. https://doi.org/10.1007/s002200050472
E. Elizalde. Ten physical applications of spectral zeta functions. Lecture Notes Physics, vol. 855, Springer, Berlin, Germany, 2012. https://doi.org/10.1007/978-3-642-29405-1
P. Epstein. Zur Theorie allgemeiner Zetafunktionen. Math. Ann., 56:615–644, 1903. https://doi.org/10.1007/BF01444309
O. M. Fomenko. Order of the Epstein zeta-function in the critical strip. J. Math. Sci., 110:3150–3163, 2002. https://doi.org/10.1023/A:1015432614102
H. Gerges, A. Laurinčikas and R. Macaitienė. A joint limit theorem for Epstein and Hurwitz zeta-functions. Mathematics, 12(13):article no. 1922, 2024. https://doi.org/10.3390/math12131922
M. L. Glasser and I. J. Zucker. Lattice sums in theoretical chemistry. Theoretical Chemistry: Advances and Perspectives (Eds. D. Henderson / H. Eyring), 5:67– 139, 1980. https://doi.org/10.1016/B978-0-12-681905-2.50008-6
J. Hadamard. Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques. Bull. Soc. Math. France, 24:199–220, 1896. https://doi.org/10.24033/bsmf.545
[19] E. Hecke. Über Modulfunktionen und die Dirichletchen Reihen mit Eulerscher Produktentwicklung. I. Math. Ann., 114:1–28, 1937. https://doi.org/10.1007/BF01594160
[20] E. Hecke. Über Modulfunktionen und die Dirichletchen Reihen mit Eulerscher Produktentwicklung. II. Math. Ann., 114:316–351, 1937. https://doi.org/10.1007/BF01594180
A. Hurwitz. Einige Eigenschaften der Dirichlet’schen Funktionen f(s) = die bei der Bestimmung der Klassenzahlen binärer quadratischer Formen auftreten. Zeitschr. für Math. und Phys., 27:86–101, 1882. https://doi.org/10.1007/978-3-0348-4161-0_3
H. Iwaniec. Topics in Classical Automorphic Forms, Graduate Studies in Mathematics. American Mathematical Society: Providence, RI, Volume 17, 1997. https://doi.org/10.1090/gsm/017
D. Joyner. Distribution theorems of L-functions. Ditman Research Notes in Math. Longman Scientific, Harlow, 1986.
E. Kowalski. An Introduction to Probabilistic Number Theory. Cambridge Univ. Press, 2021. https://doi.org/10.1017/9781108888226
A. Laurinčikas. Limit Theorems for the Riemann Zeta-Function. Kluwer, Dordrecht, 1996. https://doi.org/10.1007/978-94-017-2091-5
A. Laurinčikas. A discrete universality theorem for the Hurwitz zeta-function. J. Number Th., 143:232–247, 2014. https://doi.org/10.1016/j.jnt.2014.04.013
A. Laurinčikas. On value distribution of certain Beurling zeta-functions. Mathematics, 12(459):1–15, 2024. https://doi.org/10.3390/axioms13040251
A. Laurinčikas and R. Garunkštis. The Lerch Zeta-Function. Kluwer, Dordrecht, 2002. https://doi.org/10.1007/978-94-017-6401-8
A. Laurinčikas and R. Macaitienė. A Bohr-Jessen type theorem for the Epstein zeta-function. Results in Math., 73(4):147–163, 2018. https://doi.org/10.1007/s00025-018-0909-3
A. Laurinčikas and R. Macaitienė. A Bohr-Jessen type theorem for the Epstein zeta-function. II. Results in Math., 75(25):1–16, 2020. https://doi.org/10.1007/s00025-019-1151-3
A. Laurinčikas and D. Šiaučiūnas. Generalized limit theorem for Mellin transform of the Riemann zeta-function. Axioms, 13(251):1–17, 2024. https://doi.org/10.3390/axioms13040251
H. L. Montgomery. Topics in Multiplicative Number Theory. Springer-Verlag, Berlin, Heidelberg, New York, 1971. https://doi.org/10.1016/j.jnt.2021.03.021
T. Nakamura and L . Pańkowski. On zeros and c-values of Epstein zeta-functions. Šiauliai Math. Semin., 8:181–195, 2013.
Yu. V. Nesterenko. Modular functions and transcendence questions. Sb. Math., 187:1319–1348, 1996. https://doi.org/10.1070/SM1996v187n09ABEH000158
B. Riemann. Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie, pp. 671–680, 1859.
J. Steuding. Value-Distribution of L-Functions. Lecture Notes Math., vol. 1877, Springer, Berlin, Heidelberg, 2007. https://doi.org/10.1007/978-3-540-44822-8