## A generalization of the Pólya-Vinogradov inequality.(English)Zbl 1355.11083

Motivated by the study of sums of Dirichlet characters, E. Dobrowolski and K. S. Williams [Proc. Am. Math. Soc. 114, No. 1, 29–35 (1992; Zbl 0743.11042)] considered the following problem. Let $$f:\mathbb Z\to\mathbb C$$ be a periodic function with period $$q$$ and mean 0, that is $$\sum_{n=1}^qf(n)=0$$, let $$A=\max|f(n)|$$, and let $$B$$ be such that the inequality $\sum_{n=1}^q\left|\sum_{k=1}^Kf(n+k)\right|^2\leq BqK$ holds for all natural numbers $$K$$. In the case where $$f$$ is a non-principal Dirichlet character modulo $$q$$, it is known that $$A=B=1$$. The problem then is to obtain corresponding estimates for incomplete sums $$|\sum_{n=M+1}^{M+N}f(n)|$$. They showed, in particular, that the upper bound $\left|\sum_{n=M+1}^{M+N}f(n)\right|\leq\frac{\sqrt B}{2\log2}\sqrt q\log q+3A\sqrt q$ certainly follows. By a more careful, but still completely elementary argument, this bound was later improved by L. Rachakonda and this reviewer [Ramanujan J. 5, No. 1, 65–71 (2001; Zbl 0991.11047)].
One aim of the paper under review is to tackle this problem. Applying Fourier analysis techniques the authors show that, for $$q\geq100$$, we have $\left|\sum_{n=M+1}^{M+N}f(n)\right|\leq\frac{\sqrt{Bq}}{\pi\sqrt2}(\log q+6)+A\sqrt q.$ As an immediate corollary this furnishes an explicit form of the Pólya-Vinogradov inequality: For any non-principal character $$\chi$$ to the modulus $$q\geq100$$, we have $\left|\sum_{n=M+1}^{M+N}\chi(n)\right|\leq\frac{\sqrt{q}}{\pi\sqrt2}(\log q+6)+\sqrt q.$
Using techniques developed for this problem but specializing the argument for characters, the authors achieve their second aim obtaining the sharpest known explicit form of the Pólya-Vinogradov inequality $\left|\sum_{n=M+1}^{M+N}\chi(n)\right|\leq c\sqrt q\log q+\sqrt q,$ as follows. If $$\chi$$ is a primitive Dirichlet character to the modulus $$q\geq1200$$ and if $$\chi$$ is even (that is, $$\chi(-1)=1$$), then the bound is valid with $$c=2/\pi^2$$. But in the case of an odd primitive character ($$\chi(-1)=-1$$), the bound is valid for $$q\geq40$$ and $$c=1/(2\pi)$$.

### MSC:

 11L40 Estimates on character sums 11A25 Arithmetic functions; related numbers; inversion formulas

### Citations:

Zbl 0743.11042; Zbl 0991.11047
Full Text:

### References:

  Bachman, G.; Rachakonda, L., On a problem of dobrowolski and Williams and the Pólya-Vinogradov inequality, Ramanujan J., 5, 65-71, (2001) · Zbl 0991.11047  Burgess, D.A., On a conjecture of norton, Acta Arith., 27, 265-267, (1975) · Zbl 0258.10018  Dobrowolski, E.; Williams, K.S., An upper bound for the sum $$∑_{n=a+1}^{a+H}f(n)$$ for a certain class of functions $$f$$, Proc. Am. Math. Soc., 114, 29-35, (1992) · Zbl 0743.11042  Frolenkov, D.A., A numerically explicit version of the Pólya-Vinogradov inequality, Mosc. J. Comb. Number Theory, 1, 25-41, (2011) · Zbl 1302.11054  Granville, A.; Soundararajan, K., Large character sums: pretentious characters and the Pólya-Vinogradov theorem, J. Am. Math. Soc., 20, 357-384, (2007) · Zbl 1210.11090  Hildebrand, A., On the constant in the Pólya-Vinogradov inequality, Can. Math. Bull., 31, 347-352, (1988) · Zbl 0612.10033  Montgomery, H.L.: Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. CBMS, vol. 84. AMS, Providence (1994) · Zbl 0814.11001  Pomerance, C.: Remarks on the Pólya-Vinogradov Inequality Integers (Proceedings of the Integers Conference, October 2009), 11A (2011), Article 19, 11p
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.