Abstract
We investigate the distribution of the Riemann zeta-function on the line Re(s) = σ. For ½ < σ ≤ 1 we obtain an upper bound on the discrepancy between the distribution of ζ (s) and that of its random model, improving results of Harman and Matsumoto. Additionally, we examine the distribution of the extreme values of ζ (s) inside of the critical strip, strengthening a previous result of the first author.
As an application of these results we obtain the first effective error term for the number of solutions to ζ (s) = a in a strip ½ < σ1 < σ2 < 1. Previously in the strip ½ < σ< 1 only an asymptotic estimate was available due to a result of Borchsenius and Jessen from 1948 and effective estimates were known only slightly to the left of the half-line, under the Riemann hypothesis (due to Selberg). In general our results are an improvement of the classical Bohr–Jessen framework and are also applicable to counting the zeros of the Epstein zeta-function
Similar content being viewed by others
References
H. Bohr, and B. Jessen, Über die werteverteilung der Riemannschen zetafunktion, erste Mitteilung, Acta Math. 54 (1930), 1–35.
H. Bohr, and B. Jessen, Über die werteverteilung der Riemannschen zetafunktion, Acta Math. 58 (1932), 1–55.
V. Borchsenius, and B. Jessen, Mean motions and values of the Riemann zeta function, Acta Math. 80 (1948), 97–166.
S. Gonek, and Y. Lee, Zero-density estimates for Epstein zeta functions, Q. J. Math. 68 (2017), 301–344.
A. Granville, and K. Soundararajan, Extreme values of | (1 + it)|, in The Riemann Zeta Function and Related Themes: Papers in Honour of Professor K. Ramachandra, Ramanujan Mathematical Society, Mysore, 2006, pp. 65–80.
C. R. Guo, On the zeros of the derivative of the Riemann zeta function, Proc. London Math. Soc. (3) 72 (1996), 28–62.
G. Harman, and K. Matsumoto, Discrepancy estimates for the value-distribution of the Riemann zeta-function. IV, J. London Math. Soc. (2) 50 (1994), 17–24.
T. Hattori, and K. Matsumoto, Large deviations of Montgomery type and its application to the theory of zeta-functions, Acta Arith. 71 (1995), 79–94.
T. Hattori, and K. Matsumoto, A limit theorem for Bohr-Jessen’s probability measures of the Riemann zeta-function, J. Reine Angew. Math. 507 (1999), 219–232.
A. E. Ingham, Mean-value theorems in the theory of the Riemann zeta-function, Proc. London Math. Soc. (2) 27 (1927), 273–300.
H. Iwaniec, and E. Kowalski, Analytic Number Theory, American Mathematical Society, Providence, RI, 2004.
B. Jessen, and A. Wintner, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc. 38 (1935), 48–88.
D. Joyner, Distribution Theorems of L-functions, Longman Scientific & Technical, Harlow, 1986.
Y. Lamzouri, The two-dimensional distribution of values of ζ (1 + it), Int. Math. Res. Not. IMRN (2008), Art. ID rnn 106, 48 pp.
Y. Lamzouri, Extreme values of arg L(1, χ), Acta Arith. 146 (2011), 335–354.
Y. Lamzouri, On the distribution of extreme values of zeta and L-functions in the strip 12 < s < χ 1, Int. Math. Res. Not. IMRN (2011), 5449–5503.
E. Landau, Über die Wurzeln der Zetafunktion, Math. Z. 20 (1924), 98–104.
Y. Lee, On the zeros of Epstein zeta functions, Forum. Math. 26 (2014), 1807–1836.
K. Matsumoto, Discrepancy estimates for the value-distribution of the Riemann zeta-function. II, in Number Theory and Combinatorics, World Scientific, Singapore, 1985, pp. 265–278.
K. Matsumoto, Discrepancy estimates for the value-distribution of the Riemann zeta-function. III, Acta Arith. 50 (1988), 315–337.
K. Matsumoto, On the magnitude of asymptotic probability measures of Dedekind zeta-functions and other Euler products, Acta Arith. 60 (1991), 125–147.
H. L. Montgomery, Extreme values of the riemann zeta function, Comment. Math. Helv. 52 (1977), 511–518.
A. Selberg, Old and new conjectures and results about a class of Dirichlet series, in Proceedings of the Amalfi Conference on Analytic Number Theory, University of Salerno, Salerno, 1992, pp. 367–385.
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, The Clarendon Press Oxford University Press, New York, 1986.
Kai-Man Tsang, The Distribution of the Values of the Riemann Zeta-Function, Ph.D. Thesis, Princeton University, ProQuest LLC, Ann Arbor, MI, 1984.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 320755.
The first author is supported in part by an NSERC Discovery grant.
The third author was partially supported by NSF grant DMS-1128155.
Rights and permissions
About this article
Cite this article
Lamzouri, Y., Lester, S. & Radziwiłł, M. Discrepancy bounds for the distribution of the Riemann zeta-function and applications. JAMA 139, 453–494 (2019). https://doi.org/10.1007/s11854-019-0063-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-019-0063-1