Character sums

The main Theorem of [Qiu, 1991 †Qiu, Zhuo Ming. 1991
An inequality of Vinogradov for character sums
Shandong Daxue Xuebao Ziran Kexue Ban, 26(1), 125--128.
] implies the following result.
Theorem (1991)
For χ$\chi$ a primitive character to the modulus q>1$q > 1$, we have a=M+1M+Nχ(a)4π2qlogq+0.38q+0.637q$\left|\sum\limits_{a=M+1}^{M+N}\chi(a)\right| \le \frac{4}{\pi^2}\sqrt{q}\log q+0.38\sqrt{q}+\frac{0.637}{\sqrt{q}}$.
When χ$\chi$ is not especially primitive, but is still non-principal, we have a=M+1M+Nχ(a)863π2qlogq+0.63q+1.05q$\left|\sum\limits_{a=M+1}^{M+N}\chi(a)\right| \le \frac{8\sqrt{6}}{3\pi^2}\sqrt{q}\log q+0.63\sqrt{q}+\frac{1.05}{\sqrt{q}}$.
This was improved later by [Bachman & Rachakonda, 2001 †Bachman, Gennady, & Rachakonda, Leelanand. 2001
On a problem of Dobrowolski and Williams and the Pólya-Vinogradov inequality
Ramanujan J., 5(1), 65--71.
] into the following.
Theorem (2001)
For χ$\chi$ a non-principal character to the modulus q>1$q > 1$, we have a=M+1M+Nχ(a)13log3qlogq+6.5q$\left|\sum\limits_{a=M+1}^{M+N}\chi(a)\right| \le \frac{1}{3\log 3}\sqrt{q}\log q+6.5\sqrt{q}$.
These results are superseded by [Frolenkov, 2011 †Frolenkov, D. 2011
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)
For χ$\chi$ a non-principal character to the modulus q1000$q\ge 1000$, we have a=M+1M+Nχ(a)1π2q(logq+6)+q$\left|\sum\limits_{a=M+1}^{M+N}\chi(a)\right| \le \frac{1}{\pi\sqrt{2}}\sqrt{q}(\log q+6)+\sqrt{q}$.
In the same paper they improve upon estimates of [Pomerance, 2011 †Pomerance, C. 2011
Integers (Proceedings of the Integers Conference, October 2009), 11A, Article 19, 11pp.
] and get the following.
Theorem (2013)
For χ$\chi$ a primitive character to the modulus q1200$q \ge 1200$, we have
maxM,Na=MNχ(a){2π2qlogq+q,12πqlogq+q,χ even,χ odd.
This latter estimates holds as soon as q40$q\ge40$.
In case χ$\chi$ odd, the constant 1/(2π)$1/(2\pi)$ has already been asymptotically obtained in [Landau, 1918 †Landau, E. 1918
Abschätzungen von Charaktersummen, Einheiten und Klassenzahlen
Gött. Nachr., 2, 79--97.
] and is still unsurpassed. When χ$\chi$ is odd and M=1$M=1$, the best asymptotical constant up to now is 1/(3π)$1/(3\pi)$ from Theorem 7 of [Granville & Soundararajan, 2007 †Granville, A., & Soundararajan, K. 2007
Large character sums: pretentious characters and the Pólya-Vinogradov theorem
J. Amer. Math. Soc., 20(2), 357--384 (electronic).
], In case χ$\chi$ even, we have
maxM,Na=MNχ(a)=2maxNa=1Nχ(a).
(The LHS is always less than the RHS. Equality is then easily proved). The asymptotical best constant is 23/(35π3)$23/(35\pi\sqrt{3})$ from Theorem 7 of [Granville & Soundararajan, 2007 †Granville, A., & Soundararajan, K. 2007
Large character sums: pretentious characters and the Pólya-Vinogradov theorem
J. Amer. Math. Soc., 20(2), 357--384 (electronic).
].
2. Burgess type estimates
[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