The least quadratic nonresidue in an arithmetic sequence (Q1851512)

Let \(S_H\) be the set of numbers \(a^2+b^2\), where \(H\geq 0\) is an integer, \(0\leq a\leq[\sqrt H]\), \(0\leq b\leq[\sqrt H]\) (taking into account their multiplicities); let \(n_{\min}\) be the least quadratic nonresidue in the set \(S_H\). The author proves the following result: Theorem. Assume that for \(Q\geq H\), \(H\leq p\) the following estimate holds: \[ \left| \sum_{0\leq a,b\leq [\sqrt Q]}\left(\frac {a^2+b^2}{p}\right) \right|\ll Qp^{-\delta} \] where \(\delta>0\) is arbitrary small. Then \(n_{\min}\ll H^{\frac{1} {e^{1/\pi}}+\varepsilon}\), where \(\varepsilon>0\) is arbitrarily small.