Improving the Burgess bound via Pólya-Vinogradov

Fromm, Elijah; Goldmakher, Leo

doi:10.1090/proc/14171

Improving the Burgess bound via Pólya-Vinogradov
HTML articles powered by AMS MathViewer

by Elijah Fromm and Leo Goldmakher PDF

Proc. Amer. Math. Soc. 147 (2019), 461-466 Request permission

Abstract:

We show that even mild improvements of the Pólya-Vinogradov inequality would imply significant improvements of Burgess’ bound on character sums. Our main ingredients are a lower bound on certain types of character sums (coming from works of the second author jointly with J. Bober and Y. Lamzouri) and a quantitative relationship between the mean and the logarithmic mean of a completely multiplicative function.

References

Jonathan W. Bober and Leo Goldmakher, Pólya-Vinogradov and the least quadratic nonresidue, Math. Ann. 366 (2016), no. 1-2, 853–863. MR 3552258, DOI 10.1007/s00208-015-1353-2
D. A. Burgess, The distribution of quadratic residues and non-residues, Mathematika 4 (1957), 106–112. MR 93504, DOI 10.1112/S0025579300001157
Leo Goldmakher and Youness Lamzouri, Large even order character sums, Proc. Amer. Math. Soc. 142 (2014), no. 8, 2609–2614. MR 3209316, DOI 10.1090/S0002-9939-2014-11990-3
Leo Goldmakher and Youness Lamzouri, Lower bounds on odd order character sums, Int. Math. Res. Not. IMRN 21 (2012), 5006–5013. MR 2993442, DOI 10.1093/imrn/rnr219
Andrew Granville and K. Soundararajan, Negative values of truncations to $L(1,\chi )$, Analytic number theory, Clay Math. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 2007, pp. 141–148. MR 2362198
Andrew Granville and K. Soundararajan, Large character sums: pretentious characters and the Pólya-Vinogradov theorem, J. Amer. Math. Soc. 20 (2007), no. 2, 357–384. MR 2276774, DOI 10.1090/S0894-0347-06-00536-4
H. Halberstam and H.-E. Richert, On a result of R. R. Hall, J. Number Theory 11 (1979), no. 1, 76–89. MR 527762, DOI 10.1016/0022-314X(79)90021-0
R. R. Hall and G. Tenenbaum, Effective mean value estimates for complex multiplicative functions, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 2, 337–351. MR 1113432, DOI 10.1017/S0305004100070419
C. B. Haselgrove, A disproof of a conjecture of Pólya, Mathematika 5 (1958), 141–145. MR 104638, DOI 10.1112/S0025579300001480
Adolf Hildebrand, A note on Burgess’ character sum estimate, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), no. 1, 35–37. MR 827113
H. L. Montgomery and R. C. Vaughan, Exponential sums with multiplicative coefficients, Invent. Math. 43 (1977), no. 1, 69–82. MR 457371, DOI 10.1007/BF01390204
Terence Tao (August 18, 2009), The least quadratic nonresidue, and the square root barrier, retrieved May 1, 2017, from https://terrytao.wordpress.com/