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Statement

Model (pp. 23--24). Let $T$ be a parameter. Let $Z_3,Z_4,\ldots$ be independent random variables with $Z_n=0$ whenever $n$ has a prime factor $\le T$, and, when $n$ is free of prime factors $\le T$,

$$\mathrm{Prob}(Z_n=1)=\prod_{p\le T}\Bigl(\frac{p}{p-1}\Bigr)\cdot\frac1{\log n}, \qquad \mathrm{Prob}(Z_n=0)=1-\prod_{p\le T}\Bigl(\frac{p}{p-1}\Bigr)\cdot\frac1{\log n}.$$

For $T=1$ this is Cramér's model; Granville takes $T$ to be at least some power of $\log x$. He notes that, unlike Cramér's model, it recognizes that one of $p$ and $p+1$ is even, and it leads to the Hardy--Littlewood twin prime count (12).

Inconsistency (p. 24). Granville checks the model against Cramér's prediction (17) for primes in $(x,x+y]$ with $y/\log^2x\to\infty$ and finds, for $T=y^{1/2+o(1)}$ and $x$ divisible by $\prod_{p\le T}p$, a discrepancy by the factor $2e^{-\gamma}$ coming from Mertens's product (1); he identifies it with the inconsistency between (6) and (2) that Maier exploited.

Heuristic (p. 24, unnumbered). With the new model, Granville writes, Cramér's arguments suggest

$$\max_{p_n\le x}\,(p_{n+1}-p_n)\gtrsim 2e^{-\gamma}\log^2x,$$

which contradicts Cramér's conjecture (14); here $2e^{-\gamma}\approx1.12292$ (p. 13). He adds that the computational evidence alone would not suggest that (14) errs on the small side, but that the data are very limited.

Nothing here is proved: the display is a suggestion from a probabilistic model, and the paper gives no derivation of it beyond the reference to Cramér's argument.

Source. A. Granville, Harald Cramér and the distribution of prime numbers, Scand. Actuar. J. 1995, no. 1, 12--28: pp. 23--24, with the constant from (1) and (2) on p. 13. The edition read is identified on the source card.

Read depth. Claims checked: the model and the displayed suggestion were read clause by clause on the printed pages. Nothing here is independently reviewed.

Proof pointer

None; the paper states the suggestion without an argument written out.

Dependencies

Cramér's model and (14); Mertens's product (1), p. 13.

Bears on

• Problem 680: the paper does not mention the problem or the least prime factor of $n+k$. The heuristic predicts prime gaps larger than (14) by a factor of about $2e^{-\gamma}$; it proves nothing about either question of the problem.