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Statement

Conjecture C (printed p. 1713). "$\pi(x+y)-\pi(y)\le2\pi(x/2)$." The paper says Erdős (its reference [1], Recent Progress in Analytic Number Theory, Vol. 1, 1981) stated it as a weaker replacement for Conjecture B, and that it implies the centered interval $(-x/2,x/2)$ holds more primes than any other interval of length $x$ (p. 1714).

Table 5 (§ 4, printed p. 1718). For eleven lengths $x$ the table lists a cutoff $s$, $2\pi(x/2)$, the number $S(x)$ of elements of an admissible sequence in an interval of length $x$ built by the authors' sieve, and $2\pi(x/2)-S(x)$. In the last three rows the difference is negative:

So $\varrho^*(x)\ge S(x)>2\pi(x/2)$ at these three lengths, and the paper concludes that the last three lines "indicate the incompatibility of Conjecture C and the prime $k$-tuples conjecture" (p. 1718). In the first eight rows, $x$ from $1355252$ to $109865792$, $S(x)<2\pi(x/2)$.

The motivation (p. 1717) is Schinzel's bound $\varrho^*(x)-\pi(x)\ge(2\log2-\epsilon)x/\log^2x$, proved "assuming a special sifting hypothesis", set against $2\pi(x/2)-\pi(x)\sim\log2\times x/\log^2x$; the paper takes neither as its own result.

Source. David A. Clark and Norman C. Jarvis, "Dense admissible sequences," Mathematics of Computation 70(236) (2001), 1713--1718, https://doi.org/10.1090/s0025-5718-01-01348-5; Conjecture C on p. 1713, § 4 on pp. 1717--1718, Table 5 on p. 1718. The edition read is identified on the source card.

Read depth. Claims checked: Conjecture C, the description of the sieve and Table 5 were read on the page images of pp. 1713, 1717 and 1718, and the three negative differences were checked against the other columns. The sequences themselves, which the paper places in its ftp directory, were not examined, and nothing here is independently reviewed.

Proof pointer

§ 4, pp. 1717--1718, following Schinzel's method. Number the odd integers of the interval $n_0,n_1,\ldots$; for every prime $p$ up to the cutoff $s$ erase the class of the $n_i$ with $i\equiv-1\pmod p$, then for each later prime erase the class removing the fewest surviving elements, as in § 3. $S(x)$ is the number of survivors. The run for $x=130808636$ took about eleven days.

Dependencies

None beyond the computation; Schinzel's conditional bound (the paper's [5]) is motivation only.

Bears on

• Problem 855: Conjecture C is a weakening of the problem's inequality, with $2\pi(x/2)$ in place of $\pi(x)$. Under the prime $k$-tuples conjecture, the table's three lengths each give infinitely many $y$ with $\pi(x+y)-\pi(y)>2\pi(x/2)$, so even the weaker bound fails at those fixed $x$. Unconditionally the table says nothing about primes.