Cramer 1936 order magnitude difference between consecutive prime

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Cramér, Harald, On the order of magnitude of the difference between consecutive prime numbers. Acta Arithmetica 2 (1936), 23--46. No copyright or license line is printed on the scan's first or last page (the © mark in the ICM logo is the hosting library's watermark); the publisher's record offers the PDF under the link "Pobierz zgodnie z CC-BY" ("Free download under CC-BY license" on the English site), a Creative Commons Attribution license with no version or URL named (https://www.impan.pl/get/doi/10.4064/aa-2-1-23-46, read 2026-10-02); the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.

Cramer sets out the state of the prime-gap problem, contrasting Hoheisel's p_{n+1} - p_n = O(p_n^{1-delta}) with Westzynthius's proof that p_{n+1} - p_n = O(log p_n) is false, and Cramer's own O(sqrt(p_n) log p_n) under the Riemann hypothesis. Section 1 develops a heuristic model in which the primes are treated as one realization of a random sequence, and suggests the true maximal order of p_{n+1} - p_n is (log p_n)^2, i.e. the conjecture p_{n+1} - p_n = O((log p_n)^2) now called Cramer's conjecture. Section 2 proves unconditional and Riemann-hypothesis-conditional theorems showing that primes with exceptionally large following gaps are rare: on RH the restricted sum S_1(x) of gaps exceeding (log p_n)^3 satisfies S_1(x) = O(x / log log x) = o(x), while the full sum S(x) is asymptotic to x, a special case of the paper's Theorem II bounding the frequency of prime intervals with p_{n+1} - p_n > p_n^a (log p_n)^b. He also notes that if the conjectured (log p)^2 bound held, the series sum (p_{n+1}-p_n)^2 / (p_n (log p_n)^lambda) would converge for lambda > 4, and shows (Theorem III(b)) that this convergence for lambda > 4 holds under RH, while for lambda <= 2 the series diverges. The proofs rest on a set of lemmas, Lemma 3 among them independent of RH and yielding Hoheisel's theorem. After Theorem III (printed p. 45) he derives on RH that sum_{p_n <= x} (p_{n+1} - p_n)^2 = O(x (log x)^{3+epsilon}) for every epsilon > 0, the conditional bound that Problem 233 on the sum of the squared prime gaps asks to improve.

Source: https://matwbn.icm.edu.pl/ksiazki/aa/aa2/aa212.pdf.

Bears on. #233

Results to transcribe.

• Conjecture (4): Heuristic conjecture from the probabilistic model: p_{n+1} - p_n = O((log p_n)^2).
• Theorem II: Under the Riemann hypothesis, an upper bound for the frequency of prime intervals with p_{n+1} - p_n > p_n^a (log p_n)^b, for 0 <= a <= 1/2 and b >= 0; in particular the sum of gaps exceeding (log p_n)^3 over p_n <= x is O(x / log log x) = o(x).
• Theorem III(b): The series sum (p_{n+1}-p_n)^2 / (p_n (log p_n)^lambda) diverges for lambda <= 2, and on the Riemann hypothesis it converges for lambda > 4.
• Square sum (printed p. 45, from (57)): On the Riemann hypothesis, sum_{p_n <= x} (p_{n+1} - p_n)^2 = O(x (log x)^{3+epsilon}) for every epsilon > 0.