Quadratic Non-residues in Short Intervals

Sergei V. Konyagin; Igor E. Shparlinski

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arXiv · 1311.7016

Quadratic Non-residues in Short Intervals

Sergei V. Konyagin·Igor E. Shparlinski

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Abstract

We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes $p$ in a dyadic interval $[Q,2Q]$ for which a given interval $[u+1,u+ψ(Q)]$ does not contain a quadratic non-residue modulo $p$. The bound is nontrivial for any function $ψ(Q)\to\infty$ as $Q\to\infty$. This is an analogue of the well known estimates on the smallest quadratic non-residue modulo $p$ on average over primes $p$, which corresponds to the choice $u=0$.

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BibTeX RIS

Sergei V. Konyagin, Igor E. Shparlinski. 2013-11-27. Quadratic Non-residues in Short Intervals. https://arxiv.org/abs/1311.7016

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