Pólya-Vinogradov and the least quadratic nonresidue (Q330885)

The authors show how bounds on long character sums (in contrast to short character sums) can be used to get strong and improved bounds on the least quadratic nonresidue mod \(p\). For a given character \(\chi\pmod q\) let \(n_\chi\) be the minimal positive integer such that \(\chi(n)\neq 0,1\). Denote by \(f\) a function with nonnegative derivative (this function measures the gain in the Pólya-Vinogradov inequality). The main result states that whenever \(\max_{t\leq q}|\sum_{n\leq t}\chi(n)|\leq\sqrt{q}\cdot\frac{\log q}{f(q)}\) for all primitive even characters \(\chi\pmod q\) of order \(g\), then for all odd primitive characters \(\xi\mod k\) of order \(g\) with \(k\) not divisible by 3 one has \(n_\xi\ll\exp\left(\frac{\pi\sqrt{3}}{2(\sqrt{e}-1)}\cdot \frac{\log k}{f(k)}\right)\). In this interesting paper, the authors give several consequences and discuss also the limitations of their main result.