CaltechAUTHORS
  A Caltech Library Service

Gaps between zeros of ζ(s) and the distribution of zeros of ζ′(s)

Radziwiłł, Maksym (2014) Gaps between zeros of ζ(s) and the distribution of zeros of ζ′(s). Advances in Mathematics, 257 . pp. 6-24. ISSN 0001-8708. http://resolver.caltech.edu/CaltechAUTHORS:20180612-085638847

[img] PDF - Submitted Version
See Usage Policy.

207Kb

Use this Persistent URL to link to this item: http://resolver.caltech.edu/CaltechAUTHORS:20180612-085638847

Abstract

We assume the Riemann Hypothesis in this paper. We settle a conjecture of Farmer and Ki in a stronger form. Roughly speaking we show that there is a positive proportion of small gaps between consecutive zeros of the zeta-function ζ(s) if and only if there is a positive proportion of zeros of ζ′(s) lying very closely to the half-line. Our work has applications to the Siegel zero problem. We provide a criterion for the non-existence of the Siegel zero, solely in terms of the distribution of the zeros of ζ′(s). Finally on the Riemann Hypothesis and the Pair Correlation Conjecture we obtain near optimal bounds for the number of zeros of ζ′(s) lying very closely to the half-line. Such bounds are relevant to a deeper understanding of Levinson's method, allowing us to place one-third of the zeros of the Riemann zeta-function on the half-line.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.aim.2014.02.010DOIArticle
https://arxiv.org/abs/1301.3232arXivDiscussion Paper
Additional Information:© 2014 Elsevier Inc. Received 16 January 2013, Accepted 11 February 2014, Available online 4 March 2014. The author is partially supported by an NSERC PGS-Daward. I would like to thank Prof. Farmer, Prof. Ki and Prof. Soundararajan for comments on an early draft of this paper. I would also like to thank the anonymous referee for a very careful reading of the paper which improved the overall clarity of the paper.
Funders:
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Subject Keywords:Riemann zeta function; Zeros of the Riemann zeta-function; Zeros of the derivative of the Riemann zeta-function; Levinson's method; Siegel zeros; Class number; Pair Correlation Conjecture; Riemann Hypothesis
Classification Code:MSC: primary 11M06; secondary 11M26
Record Number:CaltechAUTHORS:20180612-085638847
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20180612-085638847
Official Citation:Maksym Radziwiłł, Gaps between zeros of ζ(s) and the distribution of zeros of ζ′(s), Advances in Mathematics, Volume 257, 2014, Pages 6-24, ISSN 0001-8708, https://doi.org/10.1016/j.aim.2014.02.010. (http://www.sciencedirect.com/science/article/pii/S0001870814000607)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:86990
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:12 Jun 2018 16:11
Last Modified:12 Jun 2018 16:11

Repository Staff Only: item control page