On the maximum number of consecutive integers on which a character is constant

Let χ be a non-principal Dirichlet character to the prime modulus p. In 1963, Burgess showed that the maximum number of consecutive integers H for which χ remains constant is O ( p1/4 log p ) . This is the best known asymptotic upper bound on this quantity. Recently, McGown proved an explicit version of Burgess’s theorem, namely that H < 7.06p1/4 log p for p ≥ 5 ·1018. Preobrazhenskaya, in the Legendre symbol case, showed that H < ( π √ 6 log 2 + o(1) ) p1/4 log p. By improving an inequality of Burgess on character sums and using some ideas of Norton, we prove that H < 1.55p1/4 log p whenever p ≥ 2.5 · 109, and H < 3.64p1/4 log p for all p.