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A Pólya–Vinogradov inequality for short character sums

Part of: Exponential sums and character sums

Published online by Cambridge University Press:  02 December 2020

Matteo Bordignon
Matteo Bordignon*
Affiliation:
School of Science, University of New South Wales Canberra, Canberra, Australia
*
e-mail:m.bordignon@student.unsw.edu.au
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Abstract

In this paper, we obtain a variation of the Pólya–Vinogradov inequality with the sum restricted to a certain height. Assume $\chi $ to be a primitive character modulo q, $ \epsilon>0$ and $N\le q^{1-\gamma }$ , with $0\le \gamma \le 1/3$ . We prove that

$$ \begin{align*} |\sum_{n=1}^N \chi(n) |\le c (\tfrac{1}{3} -\gamma+\epsilon )\sqrt{q}\log q \end{align*} $$
with $c=2/\pi ^2$ if $\chi $ is even and $c=1/\pi $ if $\chi $ is odd. The result is based on the work of Hildebrand and Kerr.

Keywords

Character sumsPólya-Vinogradov inequality

MSC classification

Primary: 11L05: Gauss and Kloosterman sums; generalizations
Secondary: 11L40: Estimates on character sums
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 906 - 910
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Copyright
© Canadian Mathematical Society 2020

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References

Bober, J., Goldmakher, L., Granville, A., and Koukoulopoulos, D., The frequency and the structure of large character sums . J. Eur. Math. Soc. (JEMS) 20(2018), 1759–1818. https://doi.org/10.4171/JEMS/799 CrossRefGoogle Scholar
Bordignon, M., Partial Gaussian sums and the Pólya–Vinogradov inequality for primitive characters. Preprint, 2020. arXiv:2001.05114 CrossRefGoogle Scholar
Bordignon, M. and Kerr, B., An explicit Pólya–Vinogradov inequality via Partial Gaussian sums . Trans. Amer. Math. Soc. 373(2020), 6503–6527. https://doi.org/10.1090/tran/8138 CrossRefGoogle Scholar
Kerr, B., On the constant in the Pólya-Vinogradov inequality . J. Number Theory 212(2020), 265–284. https://doi.org/10.1016/j.jnt.2019.11.003 CrossRefGoogle Scholar
Frolenkov, D. A. and Soundararajan, K., A generalization of the Pólya-Vinogradov inequality . Ramanujan J. 31(2013), 271–279. https://doi.org/10.1007/s11139-012-9462-y CrossRefGoogle Scholar
Granville, A. and Soundararajan, K., Large character sums: pretentious characters and the Pólya-Vinogradov theorem . J. Amer. Math. Soc. 20(2007), 357–384. https://doi.org/10.1090/S0894-0347-06-00536-4 CrossRefGoogle Scholar
Hildebrand, A., On the constant in the Pólya-Vinogradov inequality . Canad. Math. Bull. 31(1988), 347–352. https://doi.org/10.4153/CMB-1988-050-1 Google Scholar
Hildebrand, A., Large values of character sums . J. Number Theory 29(1988), 271–296. https://doi.org/10.1016/0022-314X(88)90106-0 CrossRefGoogle Scholar
Pólya, G. and Szegö, G., Problems and theorems in analysis. II. Springer-Verlag, Berlin, Heidelberg, 1998.CrossRefGoogle Scholar
Pomerance, C., Remarks on the Pólya-Vinogradov inequality . Integers 11(2011), 531–542. https://doi.org/10.1515/integ.2011.039 CrossRefGoogle Scholar
Young, W. H., On a certain series of Fourier . Proc. Lond. Math. Soc. (2) 11(1913), 357–366. https://doi.org/10.1112/plms/s2-11.1.357 CrossRefGoogle Scholar