Abstract
In this note, we present results for distinguishing L-functions by their multisets of zeros and unique factorizations in an axiomatic setting; our tools stem from universality theory.
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04 November 2021
A Correction to this paper has been published: https://doi.org/10.1007/s10986-021-09532-x
References
T.M. Apostol, Introduction to Analytic Number Theory, Springer, New York, 1976.
B. Bagchi, Recurrence in topological dynamics and the Riemann hypothesis, Acta Math. Hung., 50:227–240, 1987.
S. Bochner, On Riemann’s functional equation with multiple gamma factors, Ann. Math., 67:29–41, 1958.
H. Bohr, Über eine quasi-periodische Eigenschaft Dirichletscher Reihen mit Anwendung auf die Dirichletschen L-Funktionen, Math. Ann., 85:115–122, 1922.
E. Bombieri and A. Perelli, Distinct zeros of L-functions, Acta Arith., 83:271–281, 1998.
B. Conrey and A. Ghosh, On the Selberg class of Dirichlet series: Small degrees, Duke Math. J., 72:673–693, 1993.
P. Drungilas, R. Garunkštis, and A. Kačėnas, Universality of the Selberg zeta-function for the modular group, Forum Math., 25:533–564, 2013.
R. Garunkštis, J. Grahl, and J. Steuding, Uniqueness theorems for L-functions, Comment. Math. Univ. St. Pauli, 60:15–35, 2011.
S. Gonek, J. Haan, and H. Ki, A uniqueness theorem for functions in the extended Selberg class, Math. Z., 278:995–1004, 2014.
E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. I, Math. Z., 1: 357–376, 1918.
J. Kaczorowski, G. Molteni, A. Perelli, J. Steuding, and J. Wolfart, Hecke’s theory and the Selberg class, Funct. Approximatio, Comment. Math., 35:183–193, 2006.
J. Kaczorowski and A. Perelli, On the structure of the Selberg class. I: 0 ≤ d ≤ 1, Acta Math., 2:207–241, 1999.
J. Kaczorowski and A. Perelli, The Selberg class: a survey, in K. Győry et al. (Eds.), Number Theory in Progress, Vol. 2, de Gruyter, Berlin, 1999, pp. 953–992.
A.A. Karatsuba and S.M. Voronin, The Riemann Zeta-Function, de Gruyter, Berlin, 1992.
J. Kubilius, The distribution of Gaussian primes in sectors and contours, Uch. Zap. Leningr. Gos. Univ. Im. A.A. Zhdanova, Ser. Mat. Nauk, Tr. Astron. Obs., 137:40–52, 1950.
E. Landau, Über den Wertevorrat von ζ(s) in der Halbebeneσ > 1, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., I(36):81–91, 1933.
A. Laurinčikas, Limit Theorems for the Riemann Zeta-Function, Kluwer, Dordrecht, 1996.
A. Laurinčikas and R. Garunkštis, The Lerch Zeta-Function, Kluwer, Dordrecht, 2002.
A. Laurinčikas and K. Matsumoto, The universality of zeta-functions attached to certain cusp forms, Acta Arith., 98: 345–359, 2001.
Y. Lee, T. Nakamura, and Ł. Pańkowski, Selberg’s orthonormality conjecture and joint universality of L-functions, Math. Z., 286:1–18, 2017.
B.Q. Li, A uniqueness theorem for Dirichlet series satisfying a Riemann-type functional equation, Adv. Math., 226: 4198–4211, 2011.
J. Liu and Y. Ye, Perron’s formula and the prime number theorem for automorphic L-functions, Pure. Appl.Math. Q., 3:481–497, 2007.
K. Matsumoto, A survey on the theory of universality for zeta and L-functions, in M. Kaneko, S. Kanemitsu, and J. Liu (Eds.), Number Theory: Plowing and Starring Through HighWave Forms. Proceedings of the 7th China–Japan Seminar, Fukuoka, Japan, October 28–November 1, 2013, Ser. Number Theory Appl., Vol. 11, World Scientific, Hackensack, NJ, 2015, pp. 95–144.
H. Mishou and H. Nagoshi, The joint universality for pairs of zeta-functions in the Selberg class, Acta Math. Hung., 151:282–327, 2017.
M.R. Murty and V.K. Murty, Strong multiplicity one for Selberg’s class, C. R. Acad. Sci., Paris, Sér. I, 319:315–320, 1994.
M.R. Murty and V.K. Murty, Non-vanishing of L-Functions and Applications, Birkhäuser, Basel, 1997.
H. Nagoshi and J. Steuding, Universality for L-functions in the Selberg class, Lith. Math. J., 50(3):293–311, 2010.
N. Oswald, On a relation between modular functions and Dirichlet series: Found in the estate of Adolf Hurwitz, Arch. Hist. Exact. Sci., 71:345–361, 2017.
A. Reich, UniverselleWerteverteilung von Eulerprodukten, Nachr. Akad.Wiss. Göttingen, II. Math.-Phys. Kl., K1(1): 1–17, 1977.
Z. Rudnick and E. Waxman, Angles of Gaussian primes, Isr. J. Math., 232:159–199, 2019.
A. Selberg, Old and new conjectures and results about a class of Dirichlet series, in Collected Papers II, Springer Collect. Works Math., Springer, Berlin, Heidelberg, 1991, pp. 47–64; in E. Bombieri, A. Perelli, and U. Zannier (Eds.), Proceedings of the Amalfi Conference on Analytic Number Theory, Maiori, Italy, 25–29 September, 1989, Università di Salerno, Salerno, 1992, pp. 367–385.
K. Soundararajan, Strong multiplicity one for the Selberg class, Can. Math. Bull., 47:464–478, 2004.
K. Srinivas, Distinct zeros of functions in the Selberg class, Acta Arith., 103:201–207, 2002.
J. Steuding, Value-Distribution of L-Functions, Lect. Notes Math., Vol. 1877, Springer, Berlin, Heidelberg, 2007.
E.C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford Univ. Press, Oxford, 1939.
S. Voronin, Analytic Properties of Dirichlet Generating Functions of Arithmetic Objects, Thesis Doctor Phys.-Math. Sci., V.A. Steklov Mathematical Institute, Moscow, Russia, 1977 (in Russian).
S.M. Voronin, On the functional independence of Dirichlet L-functions, Acta Arith., 27:493–503, 1975 (in Russian).
S.M. Voronin, Theoremon the “universality” of the Riemann zeta-function, Izv. Akad. Nauk SSSR, Ser.Mat., 39:475–486, 1975 (in Russian).
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Dedicated to the Memory of Jonas Kubilius at the Occasion of his 100th Birthday
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Steuding, J. Distinguishing L-functions by joint universality. Lith Math J 61, 413–423 (2021). https://doi.org/10.1007/s10986-021-09531-y
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DOI: https://doi.org/10.1007/s10986-021-09531-y