Vinogradov's least quadratic nonresidue conjecture

Claims absolute constants $A,C>0$ with $n(p)\le C(\log p)^A$ for every odd prime $p$, hence $n(p)\ll_\delta p^\delta$ for every $\delta>0$. This is a polylogarithmic bound, much stronger than the conjecture asks, though weaker than the $(\log p)^2$ that GRH gives. The same corollary gives a deterministic polynomial-time algorithm for square roots modulo a prime. The proof is conditional on nothing beyond the family's own zero-free half-plane $\Re s>7/8$ and the published theorem of Bhargava, Ivanyos, Mittal and Saxena; it is not an independent argument.

Claims absolute constants $A,C>0$ with $n(p)\le C(\log p)^A$ for every odd prime $p$, hence $n(p)\ll_\delta p^\delta$ for every $\delta>0$. This is a polylogarithmic bound, much stronger than the conjecture asks, though weaker than the $(\log p)^2$ that GRH gives. The same corollary gives a deterministic polynomial-time algorithm for square roots modulo a prime. The proof is conditional on nothing beyond the family's own zero-free half-plane $\Re s>7/8$ and the published theorem of Bhargava, Ivanyos, Mittal and Saxena; it is not an independent argument.