Zeros of derivatives of 𝐿-functions in the Selberg class on ℜ(𝑠)<1/2

Chaubey, Sneha; Khurana, Suraj; Suriajaya, Ade

doi:10.1090/proc/16251

Zeros of derivatives of $L$-functions in the Selberg class on $\Re (s)<1/2$
HTML articles powered by AMS MathViewer

by Sneha Chaubey, Suraj Singh Khurana and Ade Irma Suriajaya;

Proc. Amer. Math. Soc. 151 (2023), 1855-1866

DOI: https://doi.org/10.1090/proc/16251

Published electronically: February 17, 2023

HTML | PDF | Request permission

Abstract:

In this article, we show that the Riemann hypothesis for an $L$-function $F$ belonging to the Selberg class implies that all the derivatives of $F$ can have at most finitely many zeros on the left of the critical line with imaginary part greater than a certain constant. This was shown for the Riemann zeta function by Levinson and Montgomery in 1974 [Acta Math. 133 (1974), pp. 49–65].

References

Hirotaka Akatsuka and Ade Irma Suriajaya, Zeros of the first derivative of Dirichlet $L$-functions, J. Number Theory 184 (2018), 300–329. MR 3724167, DOI 10.1016/j.jnt.2017.08.023
Ramūnas Garunk tis, Zeros of the extended Selberg class zeta-functions and of their derivatives, Turkish J. Math. 43 (2019), no. 6, 2921–2930. MR 4038389, DOI 10.3906/mat-1904-36
Ramūnas Garunk tis and Raivydas Šimėnas, On the Speiser equivalent for the Riemann hypothesis, Eur. J. Math. 1 (2015), no. 2, 337–350. MR 3386244, DOI 10.1007/s40879-014-0033-1
J. Hadamard, Sur la distribution des zéros de la fonction $\zeta (s)$ et ses conséquences arithmétiques, Bull. Soc. Math. France 24 (1896), 199–220 (French). MR 1504264, DOI 10.24033/bsmf.545
A. S. B. Holland, Introduction to the theory of entire functions, Pure and Applied Mathematics, Vol. 56, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 447572
Jerzy Kaczorowski, Axiomatic theory of $L$-functions: the Selberg class, Analytic number theory, Lecture Notes in Math., vol. 1891, Springer, Berlin, 2006, pp. 133–209. MR 2277660, DOI 10.1007/978-3-540-36364-4_{4}
J. Kaczorowski and A. Perelli, On the prime number theorem for the Selberg class, Arch. Math. (Basel) 80 (2003), no. 3, 255–263. MR 1981179
Li Zheng, A concise survey of the Selberg class of $L$-functions, https://personal.math.ubc.ca/~gerg/teaching/613-Winter2011/SelbergClass.pdf, 2011.
Norman Levinson and Hugh L. Montgomery, Zeros of the derivatives of the Riemann zeta-function, Acta Math. 133 (1974), 49–65. MR 417074, DOI 10.1007/BF02392141
la Vallée Poussin, Charles-Jean de, Recherches analytiques sur la théorie des nombres premiers, Annales de la Société scientifique de Bruxelles, Imprimeur de l’Académie Royale de Belgique, 20 B, 21 B: 183–256, 281–352, 363–397, 351–368.
Alberto Perelli, A survey of the Selberg class of $L$-functions. II, Riv. Mat. Univ. Parma (7) 3* (2004), 83–118. MR 2128842
Alberto Perelli, A survey of the Selberg class of $L$-functions. I, Milan J. Math. 73 (2005), 19–52. MR 2175035, DOI 10.1007/s00032-005-0037-x
Bernhard Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie, pages 671–680, 1859.
Atle Selberg, Old and new conjectures and results about a class of Dirichlet series, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989) Univ. Salerno, Salerno, 1992, pp. 367–385. MR 1220477
Raivydas Šimėnas, Zeros of the higher-order derivatives of the functions belonging to the extended Selberg class, Lith. Math. J. 61 (2021), no. 1, 87–95. MR 4230681, DOI 10.1007/s10986-020-09498-2
Rasa Šleževičienė, Speiser’s correspondence between the zeros of a function and its derivative in Selberg’s class of Dirichlet series, Fiz. Mat. Fak. Moksl. Semin. Darb. 6 (2003), 142–153 (English, with English and Lithuanian summaries). MR 2026098
Andreas Speiser, Geometrisches zur Riemannschen Zetafunktion, Math. Ann. 110 (1935), no. 1, 514–521 (German). MR 1512953, DOI 10.1007/BF01448042
Robert Spira, Zero-free regions of $\zeta ^{(k)}(s)$, J. London Math. Soc. 40 (1965), 677–682. MR 181621, DOI 10.1112/jlms/s1-40.1.677
Robert Spira, Another zero-free region for $\zeta ^{(k)}\,(s)$, Proc. Amer. Math. Soc. 26 (1970), 246–247. MR 263754, DOI 10.1090/S0002-9939-1970-0263754-4
Elias M. Stein and Rami Shakarchi, Complex analysis, Princeton Lectures in Analysis, vol. 2, Princeton University Press, Princeton, NJ, 2003. MR 1976398
Jörn Steuding, Value-distribution of $L$-functions, Lecture Notes in Mathematics, vol. 1877, Springer, Berlin, 2007. MR 2330696
Cem Yalçın Yıldırım, Zeros of derivatives of Dirichlet $L$-functions, Turkish J. Math. 20 (1996), no. 4, 521–534 (English, with English and Turkish summaries). MR 1432879