Hall 1996 proof conjecture heath brown concerning quadratic

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conjecture_p581: Heath-Brown's conjecture, which Hall proves as a corollary of his Theorem: there is an absolute delta > 0 such that for every prime p and every positive integer n, at least a proportion delta of the integers up to n are quadratic residues mod p; the paper states delta >= (1+c)/2.

inequality_15: Hall's upper bound c_0 <= -0.656999... for the limiting least mean value of a completely multiplicative f with values in [-1,1], obtained from f equal to -1 exactly on the primes in (x^{1/t}, x], whose limiting mean R(t) is smallest at t = 1 + sqrt e.

theorem_p581: Hall's Theorem that the infimum c of (1/n) sum_{m<=n} f(m), over all completely multiplicative f with -1 <= f(m) <= 1 and all n >= 1, is greater than -1; the value of c is left open.

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Hall, R. R., Proof of a conjecture of Heath-Brown concerning quadratic residues. Proc. Edinburgh Math. Soc. (2) 39 (1996), 581-588. DOI: 10.1017/S0013091500023324. The copy read for this card prints "Proceedings of the Edinburgh Mathematical Society (1996) 39, 581–588 ©" in the header of its first page, a copyright mark naming no holder, and the footer "Published online by Cambridge University Press" with the DOI on every page; the journal's Cambridge Core page was not consulted, every other right reserved.

Hall proves a conjecture Heath-Brown made informally in 1994: there is an absolute constant delta > 0 such that for all primes p and all n, at least a proportion delta of the integers up to n are quadratic residues mod p. It follows from a more general Theorem on completely multiplicative functions f with -1 <= f(m) <= 1, namely that c := inf over such f and n >= 1 of (1/n) sum_{m<=n} f(m) satisfies c > -1; applying this with the Legendre symbol (and setting f(p)=0, so multiples of p are not counted) gives delta >= (1+c)/2. The value of c, and hence the sharp delta, is left open; Section 2 (p. 584) notes that n = 3, f(2) = f(3) = -1 gives c <= -1/3 and proves c_0 <= -0.656999..., where c_0 >= c is the lim inf as x -> infinity of the infimum over f of (1/x) sum_{m<=x} f(m). The short proof relies on two deep lemmas: the sharp Hall-Tenenbaum mean-value bound with constant K = 0.32867..., where K = -cos(phi_0) for the unique root phi_0 in (0, pi) of sin phi - phi cos phi = pi/2, and a specialization of a theorem of Hildebrand involving Dickman's function; the author notes that weaker earlier mean-value results would suffice for the Theorem, and the Erdos-Ruzsa small sieve with one of them for the conjecture though not for the general Theorem.

Source: https://doi.org/10.1017/S0013091500023324.

Bears on. #121, as background only: the paper says nothing about products of integers that are squares, and its results do not bound the problem's F_k(N). The problem page cites the Theorem and the bound on c_0 as bounding the related quantity F(N) (no odd number of elements multiplying to a square), a different question from the problem's.

Results.

• Theorem (p. 581, unnumbered): the infimum c of (1/n) sum_{m<=n} f(m), over completely multiplicative f with values in [-1, 1] and all n >= 1, satisfies c > -1; its value is left open.
• Heath-Brown's conjecture (p. 581), proved as a corollary: an absolute delta > 0 bounds below the proportion of quadratic residues mod p among the integers up to n, for every prime p and every n, with delta >= (1+c)/2 as the paper states.
• Inequality (15) (p. 584): for all t > 1, R(t) >= R(1+sqrt e) = -0.656999..., where R(t) is the limiting mean of the f equal to -1 exactly on the primes in (x^{1/t}, x]; hence c_0 <= -0.656999....

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.