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The 4.36th Moment of the Riemann Zeta-Function

Radziwiłł, Maksym (2011) The 4.36th Moment of the Riemann Zeta-Function. International Mathematics Research Notices, 2012 (18). pp. 4245-4259. ISSN 1073-7928. https://resolver.caltech.edu/CaltechAUTHORS:20180614-114019101

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Abstract

Conditionally on the Riemann Hypothesis, we obtain bounds of the correct order of magnitude for the 2kth moment of the Riemann zeta-function for all positive real k<2.181. This provides for the first time an upper bound of the correct order of magnitude for some k>2; the case of k=2 corresponds to a classical result of Ingham [11]. We prove our result by establishing a connection between moments with k>2 and the so-called twisted fourth moment. This allows us to appeal to a recent result of Hughes and Young [10]. Furthermore we obtain a point-wise bound for |ζ(1/2 + it)|^(2r) (with 0<r<1) that can be regarded as a multiplicative analog of Selberg’s bound for S(T) [18]. We also establish asymptotic formulae for moments (k<2.181) slightly off the half-line.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1093/imrn/rnr183DOIArticle
http://arxiv.org/abs/1106.4806arXivDiscussion Paper
Alternate Title:The 4.36-th moment of the Riemann zeta-function
Additional Information:© The Author(s) 2011. Published by Oxford University Press. Received July 28, 2011; Accepted August 8, 2011. Advance Access Publication September 28, 2011. This work was partially supported by an NSERC PGS-D award.
Funders:
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Issue or Number:18
Record Number:CaltechAUTHORS:20180614-114019101
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20180614-114019101
Official Citation:Maksym Radziwiłł; The 4.36th Moment of the Riemann Zeta-Function, International Mathematics Research Notices, Volume 2012, Issue 18, 1 January 2012, Pages 4245–4259, https://doi.org/10.1093/imrn/rnr183
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:87102
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:14 Jun 2018 20:22
Last Modified:03 Oct 2019 19:52

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