Statistics of small prime quadratic non-residues. (English) Zbl 1511.11084
Let \( n_k(p)\) be the \(k\)-th smallest prime quadratic non-residue modulo \(p\). The author investigates the average value of \(n_k\). In this paper, the following general result is proved.
Assume that a function \(f:\mathbb{N}^k\rightarrow \mathbb{C}\) satisfies the condition \[ \max\limits_{t_1,\ldots,t_k\leqslant x}|f(t_1,\ldots,t_k)|=O\big(x^{4\sqrt{\mathrm{e}}-\varepsilon}\big) \] for some \(\varepsilon>0\). Then \[ \lim_{x\rightarrow\infty}\Big(\sum_{p\leqslant x} 1\Big)^{-1}\sum_{p\leqslant x}f(n_1(p),\ldots, n_k(p))=\hspace{-3mm} \sum_{\substack{(m_1,\ldots,m_k)\in\mathbb{N}^k \\ 1\leqslant m_1<\ldots <m_k}}\frac{f\big(p_{m_1},\ldots,p_{m_k}\big)}{2^{m_k}}. \] Several interesting conclusions from this main result are also presented.
Assume that a function \(f:\mathbb{N}^k\rightarrow \mathbb{C}\) satisfies the condition \[ \max\limits_{t_1,\ldots,t_k\leqslant x}|f(t_1,\ldots,t_k)|=O\big(x^{4\sqrt{\mathrm{e}}-\varepsilon}\big) \] for some \(\varepsilon>0\). Then \[ \lim_{x\rightarrow\infty}\Big(\sum_{p\leqslant x} 1\Big)^{-1}\sum_{p\leqslant x}f(n_1(p),\ldots, n_k(p))=\hspace{-3mm} \sum_{\substack{(m_1,\ldots,m_k)\in\mathbb{N}^k \\ 1\leqslant m_1<\ldots <m_k}}\frac{f\big(p_{m_1},\ldots,p_{m_k}\big)}{2^{m_k}}. \] Several interesting conclusions from this main result are also presented.
Reviewer: Jonas Šiaulys (Vilnius)
MSC:
| 11N60 | Distribution functions associated with additive and positive multiplicative functions |
| 11A15 | Power residues, reciprocity |
