An inequality of Vinogradov for character sums

Shandong Daxue Xuebao Ziran Kexue Ban, 26(1), 125--128.] implies the following result.

Theorem (1991)

This was improved later by [Bachman & Rachakonda, 2001 †Bachman, Gennady, & Rachakonda, Leelanand. 2001Forχ a primitive character to the modulusq>1 , we have∣∣∣∑a=M+1M+Nχ(a)∣∣∣≤4π2q√logq+0.38q√+0.637q√ .

Whenχ is not especially primitive, but is still non-principal, we have∣∣∣∑a=M+1M+Nχ(a)∣∣∣≤86√3π2q√logq+0.63q√+1.05q√ .

On a problem of Dobrowolski and Williams and the Pólya-Vinogradov inequality

Ramanujan J., 5(1), 65--71.] into the following.

Theorem (2001)

These results are superseded by [Frolenkov, 2011 †Frolenkov, D. 2011Forχ a non-principal character to the modulusq>1 , we have∣∣∣∑a=M+1M+Nχ(a)∣∣∣≤13log3q√logq+6.5q√ .

A numerically explicit version of the Pólya-Vinogradov inequality

Mosc. J. Comb. Number Theory, 1(3), 25--41.] and more recently by [Frolenkov & Soundararajan, 2013 †Frolenkov, D. A., & Soundararajan, K. 2013

A generalization of the Pólya--Vinogradov inequality

Ramanujan J., 31(3), 271--279.] into the following.

Theorem (2013)

In the same paper they improve upon estimates of [Pomerance, 2011 †Pomerance, C. 2011Forχ a non-principal character to the modulusq≥1000 , we have∣∣∣∑a=M+1M+Nχ(a)∣∣∣≤1π2√q√(logq+6)+q√ .

Remarks on the Pólya-Vinogradov inequality

Integers (Proceedings of the Integers Conference, October 2009), 11A, Article 19, 11pp.] and get the following.

Theorem (2013)

In caseForχ a primitive character to the modulusq≥1200 , we haveThis latter estimates holds as soon asmaxM,N∣∣∣∣∑a=MNχ(a)∣∣∣∣≤{2π2q√logq+q√,12πq√logq+q√,χ even,χ odd. q≥40 .

Abschätzungen von Charaktersummen, Einheiten und Klassenzahlen

Gött. Nachr., 2, 79--97.] and is still unsurpassed. When

Large character sums: pretentious characters and the Pólya-Vinogradov theorem

J. Amer. Math. Soc., 20(2), 357--384 (electronic).], In case

Large character sums: pretentious characters and the Pólya-Vinogradov theorem

J. Amer. Math. Soc., 20(2), 357--384 (electronic).]. [Booker, 2006 †Booker, A.R. 2006

Quadratic class numbers and character sums

Math. Comp., 75(255), 1481--1492 (electronic).],

Last updated on July 14th, 2012, by Olivier Ramaré