Let $p$ be an odd prime and define $n(p)$ be the smallest positive quadratic non-residue modulo $p$. By Ankeny and later effective work of Lamzouri, Li, and Soundararajan we know that under GRH one has $$n(p) \leq (\log p )^2 .$$ On the other hand under GRH, Montgomery showed that for almost all primes $p$ one has $$ n(p)\gg (\log p ) (\log \log p ).$$ My question is: what is the correct exponent? By which I mean: is there a constant $ 1\leq c < 2 $ such that firstly $$ n(p)\ll (\log p)^c $$ for all primes p, and, secondly, for all $c'>c$ there are infinitely many primes $p$ for which $$n(p)> (\log p)^{c'} .$$
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13$\begingroup$ This is an open problem, but it is generally believed (e.g. based on randomness heuristics) that Montgomery's lower bound is essentially best possible, so that $n(p) \ll_\epsilon (\log p)^{1+\epsilon}$ for all $\epsilon>0$> $\endgroup$– Thomas BloomCommented Mar 29, 2022 at 10:13
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$\begingroup$ Nice, thanks! could you please give some detail or some keywords? $\endgroup$– Dr. PiCommented Mar 30, 2022 at 7:50
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1$\begingroup$ Without appealing to GRH, Graham and Ringrose ("Lower bounds for least quadratic nonresidues", 1990) showed $n(p) \gg (\log p)(\log\log\log p)$ for infinitely many primes $p$. $\endgroup$– KConradCommented Mar 30, 2022 at 23:45
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$\begingroup$ $n(p)$ is tabulated at oeis.org/A053760 with references to the literature. $\endgroup$– Gerry MyersonCommented Mar 31, 2022 at 3:48
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