Large-sieve logarithmic exceptional-prime bound for least quadratic nonresidues ($\#\{p\le N:n(p)>p^{0.01}\}\ll \log N$)

On a dyadic interval $Q<p\le 2Q$, 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