## 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:

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