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 non-trivial for any function ψ(Q)→ ∞ as Q→ ∞. This is an analogue of the well-known estimates on the smallest quadratic nonresidue modulo p on average over primes p, which corresponds to the choice u = 0.
|Number of pages||9|
|Journal||Proceedings of the American Mathematical Society|
|Publication status||Published - 1 Oct 2015|
- Character sums
- Quadratic non-residues