Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-12-01T02:53:45.290Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
Get access

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. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.Google Scholar
[4]Granville, A. and Soundararajan, K., Large character sums. J. Amer. Math. Soc. 14(2) (2001), 365–397 (electronic).Google 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.Google 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.Google 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.Google Scholar