Baker 2001 difference between consecutive primes

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theorem_1: Baker, Harman and Pintz's short-interval theorem: every interval [x - x^{0.525}, x] with x beyond a threshold x_0 contains a prime, so consecutive primes satisfy p_{k+1} - p_k << p_k^{0.525}.

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Baker, R. C. and Harman, G. and Pintz, J., The difference between consecutive primes, II. Proc. London Math. Soc. (3) 83 (2001), 532--562. The copy read for this card is the publisher's PDF, which prints "© London Mathematical Society 2001" in the footer of its first page (the text layer renders the symbol as "q"), every other right reserved.

Theorem 1 (p. 532) proves that the closed interval [x - x^{0.525}, x] contains primes for every x > x_0, improving the previous exponent 0.535 of Baker and Harman; the authors note x_0 could in principle be made effective. The proof uses Harman's sieve method with parallel Buchstab decompositions of the sifted counts S(A, x^{1/2}) and S(B, x^{1/2}), where A is the short interval and B a long comparison interval, so that lower bounds for the short interval follow from asymptotics that hold for the long one. Instead of zero-density estimates the argument relies on mean value results for Dirichlet polynomials, in particular Watt's theorem, together with sharper estimates for six-dimensional integrals and role reversals; Lemmas 16 and 17 apply the two-dimensional sieve of §4 to obtain asymptotic formulas for sums of ordinary (one-dimensional) sifted counts. The result is the long-standing record on short intervals containing primes. Erdős problem 4's page on erdosproblems.com cites it for the best known upper bound on gaps between consecutive primes, the opposite side from the large gaps that problem asks for.

Source: https://doi.org/10.1112/plms/83.3.532.

Read status. Claims checked: Theorem 1 (p. 532), the remark on x_0 after it and the closing lower bound (p. 562) were read clause by clause on the printed pages. The proof was read for its structure only.

Bears on.

• #552: Theorem 1 implies p_{k+1} - p_k < p_k^alpha for all large k, for every alpha > 0.525 (a consequence drawn on the theorem page, not stated in the paper), which is the prime-gap hypothesis of Burr, Erdős, Faudree, Rousseau and Schelp's conditional lower bound R(C_4, K_{1,n}) > n + floor(n^{1/2} - 6n^{alpha/2}) for all large n; it sharpens the lower end of the problem's window and answers neither of its questions.
• #4, as context only: Theorem 1 implies the gap bound p_{n+1} - p_n << p_n^{0.525}, and the site cites the paper for the best known upper bound, written there as p_{n+1} - p_n << n^{0.525+o(1)}, while the problem asks for large gaps infinitely often.
• #692: Cambie's Theorem 3 on the many local maxima of the one-divisor density lists Theorem 1 among its dependencies.

Results. Theorem 1 (p. 532), with the closing quantitative bound (p. 562) and a proof pointer. Lemmas 16 and 17 (pp. 549--550) are proof steps of Theorem 1, summarized on its page; the authors say these lemmas would matter greatly for theta = 0.53 but carry little numerical weight at theta = 0.525.

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