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 \(c=1/(2\pi)\).


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