Universality of zeta-functions of cusp forms and non-trivial zeros of the Riemann zeta-function
Abstract
It is known that zeta-functions ζ(s,F) of normalized Hecke-eigen cusp forms F are universal in the Voronin sense, i.e., their shifts ζ(s + iτ,F), τ ∈ R, approximate a wide class of analytic functions. In the paper, under a weak form of the Montgomery pair correlation conjecture, it is proved that the shifts ζ(s+iγkh,F), where γ1 < γ2 < ... is a sequence of imaginary parts of non-trivial zeros of the Riemann zeta function and h > 0, also approximate a wide class of analytic functions.
How to Cite
Balčiūnas, A., Franckevič, V., Garbaliauskienė, V., Macaitienė, R., & Rimkevičienė, A. (2021). Universality of zeta-functions of cusp forms and non-trivial zeros of the Riemann zeta-function. Mathematical Modelling and Analysis, 26(1), 82-93. https://doi.org/10.3846/mma.2021.12447
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968.
R. Garunkštis and A. Laurinčikas. Discrete mean square of the Riemann zetafunction over imaginary parts of its zeros. Periodica Math. Hung., 76(2):217–228, 2018. https://doi.org/10.1007/s10998-017-0228-6
R. Garunkštis and A. Laurinčikas. The Riemann hypothesis and universality of the Riemann zeta-function. Math. Slovaca, 68(4):741–748, 2018. https://doi.org/10.1515/ms-2017-0141
R. Garunkštis, A. Laurinčikas and R. Macaitienė. Zeros of the Riemann zeta-function and its universality. Acta Arith., 181(2):127–142, 2017. https://doi.org/10.4064/aa8583-5-2017
A. Kačėnas and A. Laurinčikas. On Dirichlet series related to certain cusp forms. Lith. Math. J., 38:64–76, 1998. https://doi.org/10.1007/BF02465545
A. Laurinčikas. Non-trivial zeros of the Riemann zeta-function and joint universality theorems. J. Math. Anal. Appl., 475(1):395–402, 2019. https://doi.org/10.1016/j.jmaa.2019.02.047
A. Laurinčikas and K. Matsumoto. The universality of zeta-functions attached to certain cusp forms. Acta Arith., 98:345–359, 2001. https://doi.org/10.4064/aa98-4-2
A. Laurinčikas, K. Matsumoto and J. Steuding. Discrete universality of L-functions for new forms. Math. Notes, 78:551–558, 2003. https://doi.org//10.1007/s11006-005-0153-5
A. Laurinčikas, K. Matsumoto and J. Steuding. Discrete universality of L-functions for new forms. II. Lith. Math. J., 56:207–218, 2016. https://doi.org/10.1007/s10986-016-9314-3
A. Laurinčikas and J. Petuškinaitė. Universality of Dirichlet L-functions and non-trivial zeros of the Riemann zeta-function. Sb. Math., 210(12):1753–1773, 2019. https://doi.org/10.1070/SM9194
A. Laurinčikas, D. Šiaučiūnas and A. Vaiginytė. Extension of the discrete universality theorem for zeta-functions of certain cusp forms. Nonlinear Analysis: Modell. Control, 23(6):961–973, 2018. https://doi.org/10.15388/NA.2018.6.10
A. Laurinčikas, D. Šiaučiūnas and A. Vaiginytė. On joint approximation of analytic functions by non-linear shifts of zeta-functions of certain cusp forms. Nonlinear Analysis: Modell. Control, 25(1):108–125, 2020. https://doi.org/10.15388/namc.2020.25.15734
S.N. Mergelyan. Uniform approximations to functions of complex variable. Usp. Mat. Nauk., 7(2):31–122, 1952 (in Russian).
H.L. Montgomery. Topics in Multiplicative Number Theory. Lect. Notes Math., Vol. 227, Springer, Berlin, Heidelberg, New York, 1971. https://doi.org/10.1007/BFb0060851
H.L. Montgomery. The pair correlation of zeros of the zeta function. In H.G. Diamond(Ed.), Analytic Number Theory, volume 24 of Proc. Sympos. Pure Math., pp. 181–193. Amer. Math. Soc., Providence, RI, 1973. https://doi.org/10.1090/pspum/024/9944