Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
Hostname: page-component-78dcdb465f-tqmtl Total loading time: 14.233 Render date: 2021-04-17T15:53:25.613Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }
EnglishFrançais
Mathematika Mathematika

Article contents

THE RESONANCE METHOD FOR LARGE CHARACTER SUMS

Part of: Multiplicative number theory Exponential sums and character sums

Published online by Cambridge University Press:  11 September 2012

Bob Hough
Bob Hough
Affiliation:
Department of Mathematics, Stanford University, 450 Serra Mall, Stanford, CA 95070, U.S.A. (email: rdhough@math.stanford.edu)
Corresponding
E-mail address:
rdhough@math.stanford.edu
Get access
Rights & Permissions[Opens in a new window]

Abstract

We consider the size of large character sums, proving new lower bounds for Δ(N,q)=sup χ≠χ0 mod q∣∑ n<Nχ(n)∣ in almost all ranges of N. The proofs use the resonance method and saddle point analysis.

MSC classification

Secondary: 11L40: Estimates on character sums 11N25: Distribution of integers with specified multiplicative constraints 11N37: Asymptotic results on arithmetic functions
Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 87 - 118
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.
If you should have access and can't see this content please contact technical support.

References

[1]Burgess, D. A., The distribution of quadratic residues and non-residues. Mathematika 4 (1957), 106–112.CrossRefGoogle Scholar
[2]Farmer, D. W., Gonek, S. M. and Hughes, C. P., The maximum size of L-functions. J. Reine Angew. Math. 609 (2007), 215–236.Google Scholar
[3]Graham, S. W. and Ringrose, C. J., Lower bounds for least quadratic nonresidues. In Analytic Number Theory (Allerton Park, IL, 1989) (Progress in Mathematics 85), Birkhäuser (Boston, 1990), 269–309.CrossRefGoogle Scholar
[4]Granville, A. and Soundararajan, K., Large character sums. J. Amer. Math. Soc. 14(2) (2001), 365–397 (electronic).CrossRefGoogle Scholar
[5]Granville, A. and Soundararajan, K., Large character sums: pretentious characters and the Pólya–Vinogradov theorem. J. Amer. Math. Soc. 20(2) (2007), 357–384 (electronic).CrossRefGoogle Scholar
[6]Hildebrand, A. and Tenenbaum, G., Integers without large prime factors. J. Théor. Nombres Bordeaux 5(2) (1993), 411–484.CrossRefGoogle Scholar
[7]Hough, B., The resonance method for large character sums, arXiv:1109.1786.Google Scholar
[8]Littlewood, J. E., On the class-number of the corpus . Proc. Lond. Math. Soc. 27 (1927), 358–372.Google Scholar
[9]Milicevic, D., Large values of eigenfunctions on arithmetic hyperbolic manifolds. PhD Thesis, Princeton University, ProQuest LLC, Ann Arbor, MI, 2006.Google Scholar
[10]Montgomery, H. L. and Vaughan, R. C., Exponential sums with multiplicative coefficients. Invent. Math. 43(1) (1977), 69–82.CrossRefGoogle Scholar
[11]Ng, N., Extreme values of ζ′(ρ). J. Lond. Math. Soc. (2) 78(2) (2008), 273–289.CrossRefGoogle Scholar
[12]Soundararajan, K., Extreme values of zeta and L-functions. Math. Ann. 342(2) (2008), 467–486.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 43 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 17th April 2021. This data will be updated every 24 hours.

2
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

THE RESONANCE METHOD FOR LARGE CHARACTER SUMS
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

THE RESONANCE METHOD FOR LARGE CHARACTER SUMS
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

THE RESONANCE METHOD FOR LARGE CHARACTER SUMS
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *