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Statement

Conjecture (p. 581, quoted). "There exists an absolute positive constant $\delta$ such that for all primes $p$ and positive integers $n$, the proportion of the integers not exceeding $n$ which are quadratic residues (mod $p$) is at least $\delta$."

The paper attributes the conjecture to Roger Heath-Brown, who made it informally at the British Mathematical Colloquium in Cardiff, 1994, and proves it (p. 581) as a corollary of its Theorem, without determining the best possible $\delta$.

The deduction (p. 581). The paper applies the Theorem with the Legendre symbol $f(m)=\left(\frac mp\right)$ and states that this yields $\delta\ge(1+c)/2$, where $c>-1$ is the infimum of the Theorem. Since $f$ must be completely multiplicative, $f(p)=0$: multiples of $p$ are not counted as quadratic residues.

The paper also remarks (p. 582) that the conjecture, but not the Theorem, could be obtained from a small-sieve result of Erdős and Ruzsa together with one of the weaker versions of its Lemma 1.

Source. R. R. Hall, Proof of a conjecture of Heath-Brown concerning quadratic residues, Proc. Edinburgh Math. Soc. (2) 39 (1996), 581-588, doi:10.1017/S0013091500023324: Section 1, pp. 581-582. The edition read is identified on the source card.

Read depth. Claims checked: the conjecture and the deduction were read clause by clause on the printed page. Nothing here is independently reviewed.

Proof pointer

P. 581: the one-line application of the Theorem to the Legendre symbol described above.

Dependencies

The Theorem of the same paper.

Bears on

No Erdős problem is recorded for this statement.