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The distribution of quadratic residues and non-residues. (English) Zbl 0081.27101

The author proves the following:
Theorem 1. Let \(\delta\) and \(\varepsilon\) be any fixed positive numbers. Then, for all sufficiently large \(p\) and any \(N\), we have
\[ \left\vert \sum_{n=N+1}^{N+H} \left(\frac{n}{p}\right)\right\vert < \varepsilon H \]
provided \(H > p^{1/4+\delta}\). \(\left(\frac{n}{p}\right)\) is the Legendre symbol and \(p\) denotes a prime.
Theorem 1 immediately implies that the maximum number of consecutive quadratic residues or nonresidues \(\bmod p\) is \(O(p^{1/4+\delta})\). This is an improvement over the previous \(O(p^{1/2})\) [cf. H. Davenport and P. Erdős [Publ. Math. 2, 252–265 (1953; Zbl 0050.04302)]. With the aid of Theorem 1, the author improves on Vinogradov’s estimate \(O\left(p^{1/2\sqrt{\varepsilon} + \delta}\right)\) [Trans. Am. Math. Soc. 29, 209–217 (1927; JFM 53.0124.03)] on the least positive quadratic nonresidue \(\pmod p\). He uses Theorem 1 in place of the inequality
\[ \left\vert \sum_{n=N+1}^{N+H} \left(\frac{n}{p}\right)\right\vert < p^{1/2}\log p \]
in Vinogradov’s method to prove
Theorem 2. Let \(d\) denote the least positive quadratic non-residue \(\pmod p\). Then
\[ d = O(p^\alpha) \quad\text{as }p\to\infty,\text{ for any fixed }\alpha > \tfrac14 e^{-1/2}. \]
The proof of Theorem 1 depends on an estimate (lemma 2) which is a consequence of A. Weil’s proof of the Riemann Hypothesis for algebraic function-fields over a finite field [Sur les courbes algébriques et les variétés qui s’en déduisent. Actualités Scientifiques et Industrielles. No. 1041. Paris: Hermann (1945; Zbl 0036.16001), Deuxieme partie, §IV].

MSC:

11N69 Distribution of integers in special residue classes
11A15 Power residues, reciprocity
11L40 Estimates on character sums
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