= Large-sieve logarithmic exceptional-prime bound for least quadratic nonresidues
{title2=$\#\{p\leq N:n(p)>p^{0.01}\}\ll\log N$}
On a dyadic interval $Q<p\leq2Q$, every such exceptional prime makes all $Q^{0.01}$-smooth integers quadratic residues. By <positive density of power-smooth integers>, there are $\gg Q^2$ of these integers up to $Q^2$. Apply the <variance form of the large sieve> or its forbidden-residue consequence with interval length $Q^2$ and prime cutoff $2Q$. Each exceptional prime excludes $(p-1)/2$ residue classes and contributes at least a fixed positive amount to the sieve denominator. Hence each dyadic interval contains $O(1)$ exceptional primes. Summing the intervals gives the logarithmic bound.
// Target: analytic-number-theory.bigb
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