Stadlmann 2022 mean square gap between primes

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theorem_1: Stadlmann's unconditional bound for the mean square gap between primes: for every eps > 0 the sum over p_n <= x of (p_{n+1} - p_n)^2 is O_eps(x^(1.23 + eps)), lowering the exponent 1.25 of Peck and of Maynard.

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Julia Stadlmann, On the mean square gap between primes. arXiv preprint (2022). arXiv:2212.10867. The copy read for this card is the arXiv preprint arXiv:2212.10867v1 (21 December 2022). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2212.10867), every other right reserved.

Theorem 1 proves that sum over p_n <= x of (p_{n+1} - p_n)^2 is O_eps(x^{1.23+eps}), so the average squared prime gap below x is O(x^{0.23+eps}). In the paper's notation (1.1), with exponent 1 + nu, this is nu = 0.23, improving nu = 1/4 obtained by Peck and by Maynard and the earlier values 1/3 and 5/18 of Heath-Brown (p. 1); below nu = 1/4, as the paper explains, new phenomena and a discontinuity in the estimates of Peck and Maynard appear. The proof uses a parametric version of Harman's sieve fed by large-value estimates for Dirichlet polynomials, in particular Heath-Brown's R* bound and Heath-Brown's mean value theorem for sparse Dirichlet polynomials; it is organized into Propositions 1 and 2, Lemma 1 and Proposition 3, which respectively reduce differences of sums over a short and a long interval to large-value conditions on Dirichlet polynomials, give factor-length conditions under which those hold, compare sifted sets, and construct minorants for the prime indicator function (applying Harman's sieve six separate times).

Read status: claims checked for Theorem 1, its statement and the deduction of Section 2.5 (pp. 8--9) read clause by clause on the printed pages of arXiv v1; the proofs of Propositions 1--3 and Lemma 1 are not checked.

Source: https://arxiv.org/abs/2212.10867.

Bears on.

• #852: the paper does not mention the problem. The squares of H distinct prime gaps sum to >> H^3, so Theorem 1 bounds the length of a run of distinct consecutive gaps starting below index x; this gives the unconditional upper bound h(x) << x^{0.41+eps}, which a thread post recorded on the problem page sketches and the Theorem 1 page derives. It is a power of x and decides neither of the problem's questions, which concern the scale log x.

Results. Propositions 1--3 and Lemma 1 are ingredients of the proof of Theorem 1 and have no pages of their own.

• Theorem 1 (p. 1): For any eps > 0, sum_{p_n <= x} (p_{n+1}-p_n)^2 <<_eps x^{1.23+eps}.
• Proposition 1 (pp. 5-6): For tau in [x^{0.475-eps}, x^{0.77-eps}] and a sequence (a_n) of the admissible product shape, if every associated Dirichlet polynomial configuration satisfies one of the five large-value conditions (C1), (C2), (C3), (C4.A), (C4.B), then outside a set of O(tau x^{0.23+eps/2}) integers y in [x,3x] the sum of a_n over [y, y+y/tau] differs from x^b/tau times the sum over [y, y+y/x^b] by at most x/(tau log(x)^A).
• Proposition 2 (p. 6): For tau = x^a with a in [0.475-eps, 0.77-eps], K = 2000 and J > 10^7 K, if the relative factor lengths satisfy one of three options (one factor at least chi_0(a), a subset sum in chi_1(a), or subset sums in chi_2(a) and chi_3(a)), then for x large one of the five conditions of Proposition 1 holds.
• Lemma 1 (p. 7): Combines Propositions 1 and 2: for prime sizes P_i whose factor-length sequences all satisfy one of those options, the sifted sums over [y, y+y/tau] and x^b/tau times those over [y, y+y/x^b] agree up to (log log x)^{O(1)} y/(tau log(x)^{2+r}) for all y in [x,3x] outside a set of O(tau x^{0.23+3eps/4}) integers.
• Proposition 3 (pp. 7-8): For a in [0.475-eps, 0.77-eps] and x large there is a minorant rho <= 1_P on [x,6x], a sum of signed Buchstab terms in the ranges Lemma 1 handles, whose deficit over [y, y+y/x^b] is at most 0.99999 y/(x^b log x). Section 2.5 (pp. 8-9) combines it with Lemma 1 to show that [y, y+y/tau] contains a prime for all but O(tau x^{0.23+eps}) integers y in [x,3x], which gives Theorem 1.

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