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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)
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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

Information

Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 87 - 118
Copyright
Copyright © University College London 2012

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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 $\sqrt {-k}$. 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