Granville 1995 harald cramer distribution prime numbers

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equation_14: Granville's statement of Cramér's conjecture (14), that the largest gap between consecutive primes up to x is asymptotic to log^2 x, read off Cramér's probabilistic model, with Shanks's reformulation that the first gap exceeding g should occur near e^{(1+o(1))√g} and the table of record gaps up to 10^14; a conjecture, not a theorem.

heuristic_p24: Granville's modification of Cramér's model, which first discards integers with a prime factor at most T and then uses density 1/log n scaled by the product of p/(p-1) over p at most T; run through Cramér's argument it suggests the largest prime gap up to x is at least about 2e^{-gamma} log^2 x, contradicting Cramér's conjecture (14); not a theorem.

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A. Granville, Harald Cramér and the distribution of prime numbers, Scand. Actuar. J. 1995, no. 1, 12--28 (Harald Cramér Symposium), DOI 10.1080/03461238.1995.10413946.

The copy read for this card is an image-only scan of the seventeen printed pages with no text layer, 18 physical pages: p. 1 is printed p. 12, p. 2 is blank, and physical p. $n$ is printed p. $10+n$ for $n\ge3$. The identity was confirmed on the title page image (journal head "Scand. Actuarial J. 1995; 1: 12--28", title, author). The passages below were read on the page images, with a machine text recognition of the scan as a search aid. Provenance: downloaded in September 2026; the download URL was not recorded. 663,509 bytes. The scan prints "© 1995 Scandinavian University Press. ISSN 0346-1238" in the footer of its first page, read on the page image since the scan has no text layer, every other right reserved.

Read status: claims checked. Cramér's conjecture (14) and Shanks's reformulation on p. 21, the table on p. 22, and the corrected model and heuristic on pp. 23--24 were read clause by clause on the page images; the other pages were read on the page images for this digest, not clause by clause.

Contents

The paper is an expository survey with no new theorems.

• Pages 12--17: Euclid, Eratosthenes, Legendre's and Gauss's counts, Euler's product, Dirichlet, Riemann's explicit formula and the prime number theorem; p. 13 records Mertens's product (1), $\prod_{p\le y}(1-1/p)\sim e^{-\gamma}/\log y$, and the sieve guess (2) of about $2e^{-\gamma}x/\log x$ primes up to $x$, with $2e^{-\gamma}\approx1.12292\ldots$.
• Pages 18--19: gaps between primes. Hoheisel, Tchudakoff, Cramér and Baker--Harman on $p_{n+1}-p_n$; the Erdős--Rankin lower bound for large gaps, with Erdős's prize offer for improving its function; Brun's theorem and the Hardy--Littlewood conjectures (12) and (13) for twin primes and prime $k$-tuples.
• Pages 20--22: Cramér's model of independent "urns" with probability $1/\log n$, quoted in Cramér's words (introduced on p. 19 as his work of 1937), and its prediction (14), $\max_{p_n\le x}(p_{n+1}-p_n)\sim\log^2x$, "Cramér's Conjecture"; Shanks's reformulation, that the first gap of size $>g$ should occur with $p_n=e^{\{1+o(1)\}\sqrt g}$; and the table of record gaps up to $10^{14}$, whose largest ratio $(p_{n+1}-p_n)/\log^2p_n$ is $0.8177$.
• Pages 22--23: primes in short intervals; the Poisson law (15) for Cramér's model, which Gallagher deduced for the primes from a uniform form of (13); and Maier's theorem that, for any fixed $N>2$, there is $\delta_N>0$ such that $\pi(x+\log^Nx)-\pi(x)$ exceeds $(1+\delta_N)\log^{N-1}x$ for arbitrarily large $x$ and is below $(1-\delta_N)\log^{N-1}x$ for other arbitrarily large $x$.
• Pages 23--24: a corrected model that first removes integers with a prime factor $\le T$, for a parameter $T$, and then applies Gauss's density $1/\log n$ scaled by $\prod_{p\le T}p/(p-1)$; it predicts the Hardy--Littlewood twin prime count and, run through Cramér's argument, suggests

$$\max_{p_n\le x}(p_{n+1}-p_n)\gtrsim2e^{-\gamma}\log^2x,$$

which contradicts Cramér's conjecture (14). Granville notes that the computational evidence alone would not suggest that (14) errs on the small side.

• Pages 25--27: primes in arithmetic progressions, the failure of the averaged equidistribution conjecture found by Friedlander and Granville, and Balog's results on $k$-tuples on average; pp. 27--28 further reading and references.

Compiled scope

The passages on pp. 20--24 behind the result pages were read clause by clause on the page images; the rest of the survey was read on the page images for this digest but not checked line by line. Nothing here is independently reviewed.

Results.

• Equation (14), p. 21: Cramér's conjecture $\max_{p_n\le x}(p_{n+1}-p_n)\sim\log^2x$, with Shanks's reformulation and the table of record gaps (pp. 21--22).
• Heuristic, pp. 23--24: the sieve-corrected model and its suggestion $\max_{p_n\le x}(p_{n+1}-p_n)\gtrsim2e^{-\gamma}\log^2x$.

Bears on. #680: the paper does not mention the problem or the least prime factor of $n+k$, and proves nothing about either of its questions. It records Cramér's conjecture (14) with Shanks's form, in which a gap of size $g$ is first expected near $e^{(1+o(1))\sqrt g}$, the shape of the threshold $e^{(1+\epsilon)\sqrt k}$ in the problem's second question (an observation of this card), and the heuristic on p. 24 that the largest prime gap up to $x$ should be at least about $2e^{-\gamma}\log^2x$, contradicting (14).

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