An elementary proof of Pólya-Vinogradov's inequality (Q1964621)

Let \(\chi\) be a primitive character mod \(k\), where \(k\) is an integer with \(k>2\). The author improves the classical Pólya-Vinogradov inequality by proving for all positive integers \(h\) that \[ \left|\sum^h_{x=1} \chi (x)\right|\leq \frac 1\pi \sqrt k \log k+ \left(1-\frac{\log 2} \pi \right)\sqrt k+\frac 12, \] if \(\chi(-1)=1\), and \[ \left|\sum^h_{x=1} \chi(x)\right|\leq \frac 1\pi \sqrt k\log k +\sqrt k+\frac 12, \] if \(\chi(-1)=-1\). The author refines the standard method of proof of the Pólya-Vinogradov inequality to obtain his inequalities.