Banks 2023 ratios consecutive prime gaps

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Banks, William D., On ratios of consecutive prime gaps. Integers 23 (2023), Paper No. A50, 13 pp. The file prints "DOI: 10.5281/zenodo.8174512"; the journal's site states "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License" (https://math.colgate.edu/~integers/, read 2026-10-02), and the Zenodo record under that DOI, in the journal's repository community, states the license "Creative Commons Attribution 4.0 International" (https://zenodo.org/record/8174512, read 2026-10-02), the Creative Commons Attribution 4.0 license.

For d_n = p_(n+1) - p_n and fixed c >= 0, let pi_c(x) count primes p_n <= x with d_(n+1)/d_n >= c. The paper's Conjecture 1 asserts pi_c(x) = (c+1)^(-1) pi(x) + O(x (log x)^(-3/2+eps)) for every eps > 0, with the implied constant depending only on c and eps, a quantitative form of the Cramer-model prediction that the proportion of n with d_(n+1)/d_n >= c is 1/(c+1). The bulk of the paper is a heuristic derivation of this estimate from a strong quantitative Hardy-Littlewood prime k-tuple conjecture, in the style of Lemke Oliver and Soundararajan's work on consecutive primes in residue classes, using the Montgomery-Soundararajan refinement of the singular-series average: pi_c(x) is split into four sums F_1 through F_4, of which F_1 and the three pieces of F_2 each contribute (4c+4)^(-1) pi(x) up to smaller errors, while F_3 contributes O(x (log x)^(-3/2+eps)) and F_4 contributes O(x (log x)^(-2)). No unconditional theorem about ratios of consecutive gaps is proved; the contribution is the conjecture and its supporting computation. For problem 218, which concerns how often d_(n+1) exceeds or falls below d_n, this predicts the exact density 1/2 for d_(n+1) >= d_n and the full one-parameter family of densities.

Source: https://math.colgate.edu/~integers/vol23.html.

Bears on. #218

Results to transcribe.

• Conjecture 1: For any c >= 0 and eps > 0, pi_c(x) = (c+1)^(-1) pi(x) + O(x (log x)^(-3/2+eps)), where pi_c(x) counts p_n <= x with d_(n+1)/d_n >= c and the implied constant depends only on c and eps.
• Heuristic derivation: Conjecture 1 is derived from a strong form of the Hardy-Littlewood k-tuple conjecture together with the Montgomery-Soundararajan singular-series estimate, by splitting the count into four sums: F_1 and the three pieces G_1, G_2, G_3 of F_2 each contribute (4c+4)^(-1) pi(x), with errors at most O(x log log x (log x)^(-2)), giving the main term (c+1)^(-1) pi(x), while F_3 contributes O(x (log x)^(-3/2+eps)) and F_4 contributes O(x (log x)^(-2)) (estimates (18)-(24)).