Gaussian phenomena for small quadratic residues and non-residues. (English) Zbl 1528.11074
Let \(p\) be an odd prime and \(n_p\) denote the least quadratic non-residue modulo \(p\). I. M. Vinogradov showed that \(n_p\ll p^{\frac{1}{2\sqrt{e}}}\log p\) and conjectured that \(n_p\ll_{\varepsilon} p^{\varepsilon}\) for any fixed \(\varepsilon>0\). The best upper bound \(n_p\ll p^{\frac{1}{4\sqrt{e}}+\varepsilon}\) is due to D. A. Burgess [Mathematika 4, 106–112 (1957; Zbl 0081.27101)]. Assuming the Generalized Riemann Hypothesis (GRH), N. C. Ankeny [Ann. Math. (2) 55, 65–72 (1952; Zbl 0046.04006)] showed that \(n_p\ll (\log p)^2\), and Y. Lamzouri et al. [Math. Comput. 84, No. 295, 2391–2412 (2015; Zbl 1326.11058)] obtain an explicit bound \(n_p\le (\log p)^2\).
Given an integer \(m\) and a positive integer \(h\), denote \(S_h(m,p)=\sum_{n=m+1}^{m+h}(\frac{n}{p})\). In the paper under review, the authors prove an analogue of the result due to H. Davenport and P. Erdős [Publ. Math. Debr. 2, 252–265 (1953; Zbl 0050.04302)] in very short interval of the form \([1,(\log p)^A]\) with an arbitrary \(A>1\). They show that \(S_h(m,p)\) exhibits Gaussian distribution when \(p\) is restricted to a short interval.
Theorem 1.1. Let \(\eta\in (\frac12,1]\) and \(A>1\) be arbitrary constants. For each prime \(p\), choose an integer \(h_p\) satisfying \(h_p\to\infty\) and \(\frac{\log h_p}{\log\log p}\to 0\) as \(p\to\infty\). Then there exists an \(\eta\)-strong set \(\mathcal{B}\) of primes such that for any \(\lambda\in\mathbb{R}\), \[ \lim_{\substack{p\to\infty\\ p\in\mathcal{B}}} \frac{\#\{1\le m\le (\log p)^A: S_{h_p}(m,p)\le\lambda h_p^{\frac12}\}}{(\log p)^A} = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\lambda} e^{-t^2/2} dt, \] where a subset \(\mathcal{B}\) of prime set \(\mathbb{P}\) is called “\(\eta\)-strong” if \(\#\{(\mathbb{P}\setminus\mathcal{B})\cap [X,X+X^{\eta}]\}\le\frac{X^{\eta}}{(\log X)^{1+\delta}}\) holds for all sufficiently large \(X\) and for some \(\delta>0\).
The other problem treated in the paper is the distribution of quadratic residues and non-residues modulo two distinct primes and how they are related to each other. The authors show that the distribution is also Gaussian.
Given an integer \(m\) and a positive integer \(h\), denote \(S_h(m,p)=\sum_{n=m+1}^{m+h}(\frac{n}{p})\). In the paper under review, the authors prove an analogue of the result due to H. Davenport and P. Erdős [Publ. Math. Debr. 2, 252–265 (1953; Zbl 0050.04302)] in very short interval of the form \([1,(\log p)^A]\) with an arbitrary \(A>1\). They show that \(S_h(m,p)\) exhibits Gaussian distribution when \(p\) is restricted to a short interval.
Theorem 1.1. Let \(\eta\in (\frac12,1]\) and \(A>1\) be arbitrary constants. For each prime \(p\), choose an integer \(h_p\) satisfying \(h_p\to\infty\) and \(\frac{\log h_p}{\log\log p}\to 0\) as \(p\to\infty\). Then there exists an \(\eta\)-strong set \(\mathcal{B}\) of primes such that for any \(\lambda\in\mathbb{R}\), \[ \lim_{\substack{p\to\infty\\ p\in\mathcal{B}}} \frac{\#\{1\le m\le (\log p)^A: S_{h_p}(m,p)\le\lambda h_p^{\frac12}\}}{(\log p)^A} = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\lambda} e^{-t^2/2} dt, \] where a subset \(\mathcal{B}\) of prime set \(\mathbb{P}\) is called “\(\eta\)-strong” if \(\#\{(\mathbb{P}\setminus\mathcal{B})\cap [X,X+X^{\eta}]\}\le\frac{X^{\eta}}{(\log X)^{1+\delta}}\) holds for all sufficiently large \(X\) and for some \(\delta>0\).
The other problem treated in the paper is the distribution of quadratic residues and non-residues modulo two distinct primes and how they are related to each other. The authors show that the distribution is also Gaussian.
Reviewer: Ke Gong (Kaifeng)
MSC:
| 11L40 | Estimates on character sums |
| 11N60 | Distribution functions associated with additive and positive multiplicative functions |
| 11N36 | Applications of sieve methods |
Keywords:
Gaussian distribution; quadratic residues and non-residues; primes in short intervals; square-free numbersReferences:
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