Hostname: page-component-669899f699-chc8l Total loading time: 0 Render date: 2025-04-25T13:10:24.616Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON THE ZEROS OF L-FUNCTIONS ASSOCIATED TO FIXED-ORDER DIRICHLET CHARACTERS

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

Published online by Cambridge University Press:  18 December 2023

C. C. CORRIGAN
C. C. CORRIGAN*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We use the Weyl bound for Dirichlet L-functions to derive zero-density estimates for L-functions associated to families of fixed-order Dirichlet characters. The results improve on previous bounds given by the author when $\sigma $ is sufficiently distant from the critical line.

Keywords

Dirichlet charactersDirichlet L-functionszero-density estimates

MSC classification

Primary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Secondary: 11N35: Sieves
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 2 , October 2024 , pp. 252 - 261
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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.)

Article purchase

Temporarily unavailable

References

Baier, S. and Young, M. P., ‘Mean values with cubic characters’, J. Number Theory 130 (2010), 879–903.10.1016/j.jnt.2009.11.007CrossRefGoogle Scholar
Balestrieri, F. and Rome, N., ‘Average Bateman–Horn for Kummer polynomials’, Acta Arith. 207 (2023), 315–350.10.4064/aa220921-20-2CrossRefGoogle Scholar
Bohr, H. A. and Landau, E. G. H., ‘Ein Satz über Dirichletsche Reihen mit Anwendung auf die $\zeta$ -Funktion und die $L$ -Funktionen’, Rend. Circ. Mat. Palermo (2) 37 (1914), 269–272.10.1007/BF03014823CrossRefGoogle Scholar
Bombieri, E., ‘On the large sieve’, Mathematika 12 (1965), 201–225.10.1112/S0025579300005313CrossRefGoogle Scholar
Corrigan, C. C., On the Distribution of Zeros for Families of Dirichlet $L$ -Functions, Honours Thesis (The University of New South Wales, Sydney, 2022).Google Scholar
Corrigan, C. C., ‘Mean square values of Dirichlet $L$ -functions associated to fixed-order characters’, J. Number Theory 256 (2024), 8–22.10.1016/j.jnt.2023.09.008CrossRefGoogle Scholar
Corrigan, C. C. and Zhao, L., ‘Zero density theorems for families of Dirichlet $L$ -functions’, Bull. Aust. Math. Soc. 108 (2023), 224–235.10.1017/S0004972722001617CrossRefGoogle Scholar
Davenport, H., Multiplicative Number Theory, 3rd edn, Graduate Texts in Mathematics, 74 (Springer, Berlin, 2000).Google Scholar
Gao, P. and Zhao, L., ‘Moments of central values of quartic Dirichlet $L$ -functions’, J. Number Theory 228 (2021), 342–358.10.1016/j.jnt.2021.04.021CrossRefGoogle Scholar
Heath-Brown, D. R., ‘The density of zeros of Dirichlet’s $L$ -functions’, Canad. J. Math. 31 (1979), 231–240.10.4153/CJM-1979-024-0CrossRefGoogle Scholar
Heath-Brown, D. R., ‘A mean value estimate for real character sums’, Acta Arith. 72 (1995), 235–275.10.4064/aa-72-3-235-275CrossRefGoogle Scholar
Jutila, M., ‘On mean values of Dirichlet polynomials with real characters’, Acta Arith. 27 (1975), 191–198.10.4064/aa-27-1-191-198CrossRefGoogle Scholar
Jutila, M., ‘On mean values of $L$ -functions and short character sums with real characters’, Acta Arith. 26 (1975), 405–410.10.4064/aa-26-4-405-410CrossRefGoogle Scholar
Montgomery, H. L., ‘Zeros of $L$ -functions’, Invent. Math. 8 (1969), 346–354.10.1007/BF01404638CrossRefGoogle Scholar
Montgomery, H. L., Topics in Multiplicative Number Theory, Lecture Notes in Mathematics, 227 (Springer, Berlin, 1971).10.1007/BFb0060851CrossRefGoogle Scholar
Petrow, I. and Young, M. P., ‘The Weyl bound for Dirichlet $L$ -functions of cube-free conductor’, Ann. of Math. (2) 192 (2020), 437–486.10.4007/annals.2020.192.2.3CrossRefGoogle Scholar
Petrow, I. and Young, M. P., ‘The fourth moment of Dirichlet $L$ -functions along a coset and the Weyl bound’, Duke Math. J. 172 (2023), 1879–1960.10.1215/00127094-2022-0069CrossRefGoogle Scholar
Prachar, K., Primzahlverteilung, Grundlehren der mathematischen Wissenschaften, 91 (Springer, Göttingen, 1957).Google Scholar
Selberg, A., ‘On the zeros of Riemann’s zeta-function’, Skr. Norske Vid. Akad. Oslo I(10) (1942), 1–59.Google Scholar
Vinogradov, A. I., ‘On the density hypothesis for Dirichlet $L$ -series’, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 903–934.Google Scholar