Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-19T05:22:04.398Z Has data issue: false hasContentIssue false
  • English
  • Français

CONTINUOUS LOWER BOUNDS FOR MOMENTS OF ZETA AND L-FUNCTIONS

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  15 November 2012

Maksym Radziwiłł  and
Kannan Soundararajan
Maksym Radziwiłł
Affiliation:
Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, CA 94305-2125, U.S.A. (email: [email protected])
Kannan Soundararajan
Affiliation:
Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, CA 94305-2125, U.S.A. (email: [email protected])
Get access

Abstract

We obtain lower bounds of the correct order of magnitude for the 2kth moment of the Riemann zeta function for all k≥1. Previously such lower bounds were known only for rational values of k, with the bounds depending on the height of the rational number k. Our new bounds are continuous in k, and thus extend also to the case when k is irrational. The method is a refinement of an approach of Rudnick and Soundararajan, and applies also to moments of L-functions in families.

MSC classification

Secondary: 11M06: $zeta (s)$ and $L(s, chi)$ 11M50: Relations with random matrices
Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 119 - 128
Copyright
Copyright © University College London 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Chandee, V. and Li, X., Lower bounds for small fractional moments of Dirichlet L-functions. Int. Math. Res. Not. IMRN (2012), doi:10.1093/imrn/rns185.Google Scholar
[2]Conrey, J. B. and Ghosh, A., Mean values of the Riemann zeta-function. Mathematika 31 (1984), 159–161.Google Scholar
[3]Conrey, J. B. and Gonek, S., High moments of the Riemann zeta-function. Duke Math. J. 107 (2001), 577–604.Google Scholar
[4]Heath-Brown, D. R., Fractional moments of the Riemann zeta-function. J. Lond. Math. Soc. 24 (1981), 65–78.Google Scholar
[5]Keating, J. P. and Snaith, N. C., Random matrix theory and ζ(1/2+it). Comm. Math. Phys. 214(1) (2000), 57–89.CrossRefGoogle Scholar
[6]Radziwiłł, M., The 4.36th moment of the Riemann zeta-function. Int. Math. Res. Not. IMRN 2012 (2012), 4245–4259.Google Scholar
[7]Ramachandra, K., Some remarks on the mean value of the Riemann zeta function and other Dirichlet series. I. Hardy–Ramanujan J. 1 (1978), 1–15.Google Scholar
[8]Ramachandra, K., Some remarks on the mean value of the Riemann zeta function and other Dirichlet series. II. Hardy–Ramanujan J. 3 (1980), 1–24.Google Scholar
[9]Ramachandra, K., Some remarks on the mean value of the Riemann zeta function and other Dirichlet series. Ann. Acad. Sci. Fenn. 5 (1980), 145–158.Google Scholar
[10]Ramachandra, K., Mean value of the Riemann zeta-function and other remarks. III. Hardy–Ramanujan J. 6 (1983), 1–21.Google Scholar
[11]Rudnick, Z. and Soundararajan, K., Lower bounds for moments of L-functions. Proc. Natl. Acad. Sci. USA 102(19) (2005), 6837–6838.CrossRefGoogle ScholarPubMed
[12]Rudnick, Z. and Soundararajan, K., Lower bounds for moments of L-functions: symplectic and orthogonal examples. In Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory (Proceedings of Symposia in Applied Mathematics 75), American Mathematical Society (Providence, RI, 2006), 293–303.Google Scholar
[13]Soundararajan, K., Mean-values of the Riemann zeta-function. Mathematika 42 (1995), 158–174.Google Scholar
[14]Soundararajan, K., Moments of the Riemann zeta-function. Ann. of Math. (2) 170(2) (2009), 981–993.CrossRefGoogle Scholar
[15]Titchmarsh, E. C., The Theory of the Riemann Zeta-function, 2nd edn. (ed. D. R. Heath-Brown), Oxford Science Publications (1986).Google Scholar