Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-06-06T09:10:40.936Z Has data issue: false hasContentIssue false
  • English
  • Français

ZERO DENSITY THEOREMS FOR FAMILIES OF DIRICHLET L-FUNCTIONS

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

Published online by Cambridge University Press:  13 January 2023

CHANDLER C. CORRIGAN  and
LIANGYI ZHAO Open the ORCID record for LIANGYI ZHAO [Opens in a new window]
CHANDLER C. CORRIGAN
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia e-mail: c.corrigan@student.unsw.edu.au
LIANGYI ZHAO*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
*
e-mail: l.zhao@unsw.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove some zero density theorems for certain families of Dirichlet L-functions. More specifically, the subjects of our interest are the collections of Dirichlet L-functions associated with characters to moduli from certain sparse sets and of certain fixed orders.

Keywords

Dirichlet characterszero density theorems

MSC classification

Primary: 11M20: Real zeros of $L(s, chi)$; results on $L(1, chi)$
Secondary: 11N35: Sieves
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 2 , October 2023 , pp. 224 - 235
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.)

References

Baier, S., ‘On the large sieve with sparse sets of moduli’, J. Ramanujan Math. Soc. 21 (2006), 279–295.Google Scholar
Baier, S. and Young, M. P., ‘Mean values with cubic characters’, J. Number Theory 130(4) (2010), 879–903.10.1016/j.jnt.2009.11.007CrossRefGoogle Scholar
Baier, S. and Zhao, L., ‘Large sieve inequalities for characters to powerful moduli’, Int. J. Number Theory 1(2) (2005), 265–279.10.1142/S1793042105000170CrossRefGoogle Scholar
Baier, S. and Zhao, L., ‘An improvement for the large sieve for square moduli’, J. Number Theory 128(1) (2008), 154–174.10.1016/j.jnt.2007.03.004CrossRefGoogle Scholar
Baker, R. C., Munsch, M. and Shparlinski, I. E., ‘Additive energy and a large sieve inequality for sparse sequences’, Mathematika 68 (2022), 362–399.10.1112/mtk.12140CrossRefGoogle Scholar
Dunn, A. and Radziwiłł, M., ‘Bias in cubic Gauss sums: Patterson’s conjecture’, Preprint, 2021, arXiv:2109.07463.2021 Google Scholar
Gallagher, P. X., ‘A large sieve density estimate near $\sigma =1$ ’, Invent. Math. 11 (1970), 329–339.10.1007/BF01403187CrossRefGoogle Scholar
Gao, P. and Zhao, L., ‘Large sieve inequalities for quartic characters’, Q. J. Math. 63(4) (2012), 891–917.10.1093/qmath/har018CrossRefGoogle 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
Halupczok, K., ‘Large sieve inequalities with general polynomial moduli’, Q. J. Math. 66(2) (2015), 529–545.10.1093/qmath/hav011CrossRefGoogle Scholar
Halupczok, K., ‘Bounds for discrete moments of Weyl sums and applications’, Acta Arith. 194(1) (2020), 1–28.10.4064/aa181207-23-9CrossRefGoogle Scholar
Halupczok, K. and Munsch, M., ‘Large sieve estimate for multivariate polynomial moduli and applications’, Monatsh. Math. 197(3) (2022), 463–478.10.1007/s00605-021-01641-6CrossRefGoogle Scholar
Heath-Brown, D. R., ‘A mean value estimate for real character sums’, Acta Arith. 72(3) (1995), 235–275.10.4064/aa-72-3-235-275CrossRefGoogle Scholar
Iwaniec, H. and Kowalski, E., Analytic Number Theory, American Mathematical Society Colloquium Publications, 53 (American Mathematical Society, Providence, RI, 2004).Google 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
Montgomery, H. L., Topics in Multiplicative Number Theory, Lecture Notes in Mathematics, 227 (Spring-Verlag, Berlin, 1971).10.1007/BFb0060851CrossRefGoogle Scholar
Munsch, M., ‘A large sieve inequality for power moduli’, Acta Arith. 197(2) (2021), 207–211.10.4064/aa191212-1-6CrossRefGoogle Scholar
Ramaré, O., Arithmetical Aspects of the Large Sieve Inequality, Harish-Chandra Research Institute Lecture Notes, 1 (Hindustan Book Agency, New Delhi, 2009).10.1007/978-93-86279-40-8CrossRefGoogle Scholar
Zhao, L., ‘Large sieve inequality for characters to square moduli’, Acta Arith. 112(3) (2004), 297–308.10.4064/aa112-3-5CrossRefGoogle Scholar