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For a prime $p$ denote by $r(p)$ (resp. $n(p)$) the smallest prime $q$ which is a quadratic residue (resp. nonresidue) modulo $p$. It was shown by Linnik that for any fixed $\epsilon>0$ the number of $p<x$ s.t. $n(p)>p^\epsilon$ is bounded by $c(\epsilon)\log\log x$.

My question is what is the best known bound today and what is the best known corresponding bound for $r(p)$.

Edit: I mean a bound on the number of exceptions to Vinogradov's conjectures, i.e. $n(p),r(p)>p^\epsilon$.

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2 Answers 2

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There is the following result of Wolke from $1967$ (which is perhaps not the best, but quite good). Theorem: Let $p$ be an odd prime, and $L(s,\chi)$ the $L$-series for the Dirichlet character $(n/p)$. If $t(p)$ is a positive function with $L(1,\chi)>t(p)/\log(p)$, then there are absolute constants $c_1,c_2>0$ with $$ r(p)\le c_1 p^{c_2/\sqrt{t(p)}}. $$ Here one should mention the result of Elliott: if we have for an integer $k\ge 0$ and real $c>0$ $$ L(1,\chi)\ge \frac{c(\log \log p)^k}{\log p}, $$ then for every $\epsilon >0$ we have $r(p)\le c(\epsilon)p^{1/4(1+\epsilon)(k+2)^{-1}}$.

For $n(p)$ see the the report of Terence Tao, and http://www.math.ubc.ca/~gerg/teaching/613-Winter2011/LeastQuadraticNonResidue.pdf.

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  • $\begingroup$ The result of Elliott you mentioned seems too good to be true, where can I find it? $\endgroup$
    – Alex
    Jul 25, 2013 at 13:55
  • $\begingroup$ @Alex: P.D.T.A. Elliott: A note on a recent paper of U.V. Linnik and A.I. Vinogradov, Acta Arithm. 13(1967), pp. 103-105. I had a typo in my answer here, sorry. $\endgroup$ Jul 25, 2013 at 15:01
  • $\begingroup$ Thanks for the clarification. Is there an unconditional bound on the number of $p<x$ s.t. $L(\chi_p,1)<t/\log p$ for every fixed $t$? $\endgroup$
    – Alex
    Jul 25, 2013 at 15:39
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Terry Tao has a nice blog post about this.

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