# zbMATH — the first resource for mathematics

Gram points in the theory of zeta-functions of certain cusp forms. (English) Zbl 1468.11173
Summary: In the paper, an analogue of the Gram points used in the theory of the Riemann zeta-function is introduced for zeta-functions of normalized Hecke-eigen cusp forms of weight $$\kappa$$. Some analytic properties of those points are studied, and the first ten Gram points for $$\kappa\leqslant 12$$ are calculated. The main attention is devoted to the universality of zeta-functions of cusp forms on the approximation of analytic functions by shifts involving the sequence of Gram points.
##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M41 Other Dirichlet series and zeta functions 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses
Full Text:
##### References:
 [1] Billingsley, P., Convergence of Probability Measures (1968), Willey: Willey New York · Zbl 0172.21201 [2] (Carlson, J.; Joffe, A.; Wiles, A., The Millennium Prise Problems (2006), Clay Math. Institute, Amer. Math. Soc.: Clay Math. Institute, Amer. Math. Soc. Providence, RI) [3] de Bruijn, N. G., Asymptotics Methods in Analysis (1958), North-Holland Publ. Co.: North-Holland Publ. Co. Amsterdam · Zbl 0082.04202 [4] Gram, J. P., Sur les zéros de la fonction $$\zeta(s)$$ de Riemann, Acta Math., 27, 289-304 (1903) · JFM 34.0461.01 [5] Hatchinson, J. I., On the roots of the Riemann zeta-function, Trans. Am. Math. Soc., 27, 49-60 (1925) · JFM 51.0267.01 [6] Ivič, A., The Riemann Zeta-Function. The Theory of the Riemann Zeta-Function with Applications (1985), John Wiley & Sons: John Wiley & Sons New York, Chichester, Brisbane, Toronto, Singapore · Zbl 0556.10026 [7] Kačėnas, A.; Laurinčikas, A., On Dirichlet series related to certain cusp forms, Lith. Math. J., 38, 64-76 (1998) · Zbl 0926.11064 [8] Kaczorowski, J.; Perelli, A., The Selberg class: a survey, (Number Theory in Progress. Number Theory in Progress, Proc. Intern. Conf. in Honor of the 60th Birthday of A. Schinzel, vol. 2 (1997), de Gryter: de Gryter Zakopane), 953-992 · Zbl 0929.11028 [9] Korolev, M. A., Gram’s law and Selberg’s conjecture on the distribution of the zeros of the Riemann zeta-function, Izv. Math., 74, 743-780 (2010) · Zbl 1257.11080 [10] Korolev, M. A., On Gram’s law in the theory of the Riemann zeta-function, Izv. Math., 76, 275-309 (2012) · Zbl 1250.11081 [11] Korolev, M. A., Gram’s law and the argument of the Riemann zeta-function, Publ. Inst. Math. (Belgr.) (N.S.), 92, 106, 53-78 (2012) · Zbl 1274.11132 [12] Korolev, M. A., On the Selberg formulas related to Gram’s law, Sb. Math., 203, 1808-1816 (2012) · Zbl 1321.11083 [13] Korolev, M. A., On new results related to Gram’s law, Izv. Math., 77, 917-940 (2013) · Zbl 1303.11100 [14] Korolev, M. A., On small values of the Riemann zeta-function at the Gram points, Sb. Math., 205, 63-82 (2014) · Zbl 1318.11107 [15] Korolev, M. A., Gram’s law in the theory of the Riemann zeta-function. Part 1, Proc. Steklov Inst. Math., 292, 1-146 (2016) · Zbl 1361.11051 [16] Korolev, M. A., Gram’s law in the theory of the Riemann zeta-function. Part 2, Proc. Steklov Inst. Math., 294, 1-78 (2016) · Zbl 1403.11056 [17] Korolev, M. A.; Laurinčikas, A., A new application of the Gram points, Aequ. Math., 93, 859-873 (2019) · Zbl 1470.11223 [18] Kuipers, L.; Niederrater, H., (Uniform Distribution of Sequences. Uniform Distribution of Sequences, Pure and Appl. Math. (1974), Willey: Willey New York) [19] Laurinčikas, A.; Matsumoto, K., The universality of zeta-functions attached to certain cusp forms, Acta Arith., 98, 345-359 (2001) · Zbl 0974.11018 [20] Laurinčikas, A.; Matsumoto, K.; Steuding, J., Discrete universality of L-functions for new forms, Math. Notes, 78, 551-558 (2005) · Zbl 1161.11381 [21] Laurinčikas, A.; Matsumoto, K.; Steuding, J., Discrete universality of L-functions for new forms. II, Lith. Math. J., 56, 207-218 (2016) · Zbl 1351.11057 [22] Mergelyan, S. N., Uniform approximations to functions of a complex variable, Usp. Mat. Nauk, 7, 31-122 (1952), (in Russian) · Zbl 0059.05902 [23] Montgomery, H. L., Topics in Multiplicative Number Theory, Lecture Notes Math., vol. 227 (1971), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0216.03501 [24] Olver, F. W.J., Asymptotics and Special Functions (1997), AK Peters: AK Peters Wellesley, Massachusets · Zbl 0982.41018 [25] Selberg, A., The zeta-functions and the Riemann hypothesis, (C. R. Dixiém Congrés Math. Scandinaves 1946, Jul. (1947), Gjellerups Forlag: Gjellerups Forlag Copenhagen), 187-200 [26] Selberg, A., Old and new conjectures and results about a class of Dirichlet series, (Bombieri, E.; etal., Proc. Amalfi Conf. Analytic Number Theory. Proc. Amalfi Conf. Analytic Number Theory, Maiori, 1989 (1992), Universitié di Salerno), 367-385 · Zbl 0787.11037 [27] Steuding, J., Value-Distribution of L-Functions, Lecture Notes Math., vol. 1877 (2007), Springer: Springer Berlin, Heidelberg, New York · Zbl 1130.11044 [28] Steuding, J., On the value distribution of L-functions, Fiz. Math. Fak. Moksl. Semin. Darb., 6, 87-119 (2009) · Zbl 1132.11344 [29] Titchmarsh, E. C., The zeros of the Riemann zeta-function, Proc. R. Soc. Lond. Ser. A, 151, 234-255 (1935) · Zbl 0012.29701 [30] Titchmarsh, E. C., The zeros of the Riemann zeta-function, Proc. R. Soc. Lond. Ser. A, 157, 261-263 (1936) · JFM 62.0343.03 [31] Trudgian, T. S., On the success and failure of Gram’s law and the Rosser rule, Acta Arith., 143, 225-256 (2011) · Zbl 1281.11084
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.