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Statement

$H(k)$ is the minimal diameter $h_k-h_1$ of an admissible $k$-tuple, a tuple of $k$ increasing integers avoiding at least one residue class modulo every prime (pp. 1, 9). In the section "Narrow admissible tuples", under "Sieving methods" (p. 78): the sieve of Eratosthenes, which sieves $[2,x]$ by the class $0\pmod p$ for all $p\le k$ and takes the survivors $p_{m+1},\ldots,p_{m+k}$ with $m=\pi(k)$, "yields the upper bound"

$$H(k)\le k\log k+k\log\log k-k+o(k) \qquad (149)$$

by the prime number theorem in the forms $p_k=k\log k+k\log\log k-k+O(k\log\log k/\log k)$ and $\pi(x)=x/\log x+O(x/\log^2x)$. Hensley and Richards [44--46] improved (149) by sieving the symmetric interval $[-x/2,x/2]$ in place of $[2,x]$, which gives admissible $k$-tuples made of $-1$, $1$, the primes $p_{m+1},\ldots,p_{m+\lfloor(k+1)/2\rfloor-1}$ and the negatives of $p_{m+1},\ldots,p_{m+\lfloor k/2\rfloor-1}$, again with $m$ as small as possible (the printed display of this tuple omits the minus sign of $-p_{m+1}$); "It follows from Lemma 5 of [45] that one can take $m=o(k/\log k)$, leading to the improved upper bound"

$$H(k)\le k\log k+k\log\log k-(1+\log2)k+o(k). \qquad (150)$$

Theorem 17 (vi), (xi) (p. 10) states the Eratosthenes bound as a theorem with an effective $o(k)$; the paper then describes the shifted Schinzel and shifted greedy sieves (pp. 78--79) as further numerical improvements and (p. 79) conjectures $k\log k+k$ as an upper bound for all large $k$.

Source. D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Res. Math. Sci. 1 (2014), Art. 12; displays (149)--(150) on p. 78 of the journal PDF, read in the text layer; Theorem 17 on p. 10. The arXiv version (1407.4897) was not compared; its section numbering differs from the journal's unnumbered headings.

Read depth. Claims checked: the two displays, the sentences around them and Theorem 17 (vi), (xi) were read clause by clause in the text layer. The displays come with a one-sentence justification each, and the derivation of (150) from Lemma 5 of Hensley and Richards is not carried out in the paper; nothing was checked beyond the statements.

Proof pointer

Page 78: (149) from the prime number theorem applied to $p_{\pi(k)+k}$; (150) from the symmetric sieve of $[-x/2,x/2]$, whose tuple is drawn from $\pm1$ and the $\pm p_j$ with $j>m$, where $m=o(k/\log k)$ suffices, which the paper attributes to Lemma 5 of Hensley and Richards 1974 (that paper's Theorem gives the same gain in the form $\rho^*(x)-\pi(x)\ge(\log2-\varepsilon)x/(\log x)^2$).

Dependencies

The prime number theorem with the stated error terms; Hensley and Richards's Lemma 5 (their Acta Arithmetica paper, held). References [44]--[46] of the paper are Hensley and Richards 1973 (the symposium paper), Hensley and Richards 1974 (Acta Arith. 25) and Richards 1974 (Bull. Amer. Math. Soc. 80).

Bears on

• Problem 1204: $H(k)$ is the problem's $A(k)$, and (150) is the second-order improvement of the upper bound that the site attributes to Hensley and Richards.