---

Statement

Theorem 1 (p. 532, quoted). "For all $x>x_0$, the interval $[x-x^{0.525},x]$ contains prime numbers."

The paper adds directly after the theorem (p. 532) that with enough effort $x_0$ could be determined effectively; it gives no value of $x_0$. The interval is closed. The exponent improves the $0.535$ that the paper credits to Baker and Harman (its reference [1], Proc. London Math. Soc. (3) 72 (1996), 261--280).

The quantitative form (p. 562). The proof ends, with $\theta=0.525$ and the small $\varepsilon$ of the construction suppressed "for brevity" (p. 557), in the lower bound

$$\pi(x+x^{0.525})-\pi(x)\ge\frac{9}{100}\,\frac{x^{0.525}}{\log x}$$

for all large $x$, the constant being $1$ less the bounds $0.3$ (regions $A$ and $B$), $0.06$ ($E$ and $F$), $0.21$ ($C$) and $0.34$ ($D$) that Section 6 obtains for the losses in the regions of the final decomposition (p. 561). The paper does not restate this bound as a numbered result.

Prime gaps (a consequence drawn on this page, not stated in the paper). For a large prime $p_k$ put $x=p_k+2p_k^{0.525}$; then $x-x^{0.525}>p_k$, and Theorem 1 puts a prime in $(p_k,x]$. Hence $p_{k+1}-p_k\le2p_k^{0.525}$ for all large $k$, and so $p_{k+1}-p_k<p_k^{\alpha}$ for all large $k$, for every fixed $\alpha>0.525$.

Source. R. C. Baker, G. Harman and J. Pintz, The difference between consecutive primes, II, Proc. London Math. Soc. (3) 83 (2001), 532--562, doi:10.1112/plms/83.3.532: Theorem 1 and the outline of the method in Section 1 (pp. 532--535); Section 2, the application of Watt's theorem (pp. 535--539); Section 3, sieve asymptotic formulae (pp. 539--545); Section 4, the two-dimensional sieve (pp. 545--551), with Lemma 16 (p. 549) and Lemma 17 (p. 550); Section 5, further asymptotic formulae (pp. 551--557); Section 6, the final decomposition and the closing bound (pp. 557--562). The edition read is identified on the source card.

Read depth. Claims checked: the statement, the remark on $x_0$ and the closing bound were read clause by clause on the printed pages. The proof was read for its structure only and was not checked; the numerical integrations behind the losses in Section 6 were not recomputed. Nothing here is independently reviewed.

Proof pointer

Pp. 532--562. With $\mathcal A=[x-y,x)\cap\mathbb Z$ for $y=x^{\theta+\varepsilon}$ and a long comparison interval $\mathcal B=[x-y_1,x)\cap\mathbb Z$, $y_1=x\exp(-3(\log x)^{1/3})$, the count of primes in $\mathcal A$ is the sifted count $S(\mathcal A,x^{1/2})$ (p. 533). Buchstab's identity is applied in parallel to $S(\mathcal A,x^{1/2})$ and $S(\mathcal B,x^{1/2})$: terms with an asymptotic formula transfer from $\mathcal B$ to $\mathcal A$ with the factor $y/y_1$, the remaining terms, being non-negative, are discarded where they enter with a plus sign, and the theorem follows once the discarded part is shown to be less than the whole. The asymptotic formulae come from mean value estimates for Dirichlet polynomials rather than zero-density estimates, with Watt's mean value theorem (Section 2) supplying much of the gain over the exponent $0.535$, and from a two-dimensional sieve (Section 4) that handles some sums of one-dimensionally sifted counts (Lemmas 16 and 17), together with reversals of the roles of variables. The paper says (p. 532) that Lemmas 16 and 17 and the role reversals matter little numerically at $0.525$. Section 6 sets $\theta=0.525$, splits the range of $\Sigma_3$ in (1.2) into regions $A$ to $F$, and bounds the loss from each by numerical integration.

Dependencies

The sieve method of Harman (the paper's references [4] and [5]); N. Watt, Kloosterman sums and a mean value for Dirichlet polynomials, J. Number Theory 53 (1995), 179--210 (reference [11]); and mean value results and lemmas from Baker--Harman (reference [1]), Baker--Harman--Pintz (reference [2]), Baker--Harman--Rivat (reference [3]) and Heath-Brown (references [6] and [7]), cited where used.

Bears on

• Problem 552: Burr, Erdős, Faudree, Rousseau and Schelp's Theorem 2 bounds $R(C_4,K_{1,n})$ from below under the hypothesis that $p_{k+1}-p_k<p_k^{\alpha}$ for all large $k$. By the prime-gap consequence above, Theorem 1 supplies that hypothesis for every $\alpha>0.525$, giving $R(C_4,K_{1,n})>n+\lfloor n^{1/2}-6n^{\alpha/2}\rfloor$ for all large $n$, for each such $\alpha$. This sharpens the lower end of the problem's window and does not answer either of its questions.
• Problem 4, as context only: the gap bound $p_{k+1}-p_k\ll p_k^{0.525}$ is an upper bound on gaps between consecutive primes, while the problem asks for large gaps infinitely often.
• Problem 692: Cambie's Theorem 3 page lists Theorem 1 among its dependencies, using it to place a prime in each of many short intervals of length of order $X^{0.525}$.