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THE RESONANCE METHOD FOR LARGE CHARACTER SUMS

  • Bob Hough (a1)

Abstract

We consider the size of large character sums, proving new lower bounds for Δ(N,q)=sup χχ0 mod q∣∑ n<Nχ(n)∣ in almost all ranges of N. The proofs use the resonance method and saddle point analysis.

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[1]Burgess, D. A., The distribution of quadratic residues and non-residues. Mathematika 4 (1957), 106112.
[2]Farmer, D. W., Gonek, S. M. and Hughes, C. P., The maximum size of L-functions. J. Reine Angew. Math. 609 (2007), 215236.
[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), 269309.
[4]Granville, A. and Soundararajan, K., Large character sums. J. Amer. Math. Soc. 14(2) (2001), 365397 (electronic).
[5]Granville, A. and Soundararajan, K., Large character sums: pretentious characters and the Pólya–Vinogradov theorem. J. Amer. Math. Soc. 20(2) (2007), 357384 (electronic).
[6]Hildebrand, A. and Tenenbaum, G., Integers without large prime factors. J. Théor. Nombres Bordeaux 5(2) (1993), 411484.
[7]Hough, B., The resonance method for large character sums, arXiv:1109.1786.
[8]Littlewood, J. E., On the class-number of the corpus . Proc. Lond. Math. Soc. 27 (1927), 358372.
[9]Milicevic, D., Large values of eigenfunctions on arithmetic hyperbolic manifolds. PhD Thesis, Princeton University, ProQuest LLC, Ann Arbor, MI, 2006.
[10]Montgomery, H. L. and Vaughan, R. C., Exponential sums with multiplicative coefficients. Invent. Math. 43(1) (1977), 6982.
[11]Ng, N., Extreme values of ζ′(ρ). J. Lond. Math. Soc. (2) 78(2) (2008), 273289.
[12]Soundararajan, K., Extreme values of zeta and L-functions. Math. Ann. 342(2) (2008), 467486.
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