Predicting maximal gaps in sets of primes

Alexei Kourbatov; Marek Wolf

doi:10.3390/math7050400

paper

Predicting maximal gaps in sets of primes

arXiv:1901.03785 · doi:10.3390/math7050400

Abstract

Let $q > r \geq 1$ be coprime integers. Let $P_{c} = P_{c} (q, r, H)$ be an increasing sequence of primes $p$ satisfying two conditions: (i) $p \equiv r$ (mod $q$ ) and (ii) $p$ starts a prime $k$ -tuple with a given pattern $H$ . Let $π_{c} (x)$ be the number of primes in $P_{c}$ not exceeding $x$ . We heuristically derive formulas predicting the growth trend of the maximal gap $G_{c} (x) = max_{p^{'} \leq x} (p^{'} - p)$ between successive primes $p, p^{'} \in P_{c}$ . Extensive computations for primes up to $1 0^{14}$ show that a simple trend formula $G_{c} (x) \sim \frac{x}{π _{c} ( x )} \cdot (lo g π_{c} (x) + O_{k} (1))$ works well for maximal gaps between initial primes of $k$ -tuples with $k \geq 2$ (e.g., twin primes, prime triplets, etc.) in residue class $r$ (mod $q$ ). For $k = 1$ , however, a more sophisticated formula $G_{c} (x) \sim \frac{x}{π _{c} ( x )} \cdot (lo g \frac{π _{c}^{2} ( x )}{x} + O (lo g q))$ gives a better prediction of maximal gap sizes. The latter includes the important special case of maximal gaps in the sequence of all primes ( $k = 1$ , $q = 2$ , $r = 1$ ). The distribution of appropriately rescaled maximal gaps $G_{c} (x)$ is close to the Gumbel extreme value distribution. Computations suggest that almost all maximal gaps satisfy a generalized strong form of Cramer's conjecture. We also conjecture that the number of maximal gaps between primes in $P_{c}$ below $x$ is $O_{k} (lo g x)$ .

arXiv comment30 pages, 10 figures, 1 table. Errata (last page); URL ref.[3] updated. Sequel to arXiv:1102.0481, arXiv:1610.03340, arXiv:1709.05508

Predicting maximal gaps in sets of primes

Abstract

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