Bounded gaps between primes of special form

A. V. Shubin

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 492 (2020), pp. 75-78

Résumé

Let $0\alpha$, $\sigma1$ be arbitrary fixed constants, let $q_1$ be the set of primes satisfying the condition $\{q_n^\alpha\}\sigma$ and indexed in ascending order, and let $m\ge1$ be any fixed integer. Using an analogue of the Bombieri–Vinogradov theorem for the above set of primes, upper bounds are obtained for the constants $c(m)$ such that the inequality $q_{n+m}-q_n\le c(m)$ holds for infinitely many $n$.

Détail

Keywords: consecutive primes, small gaps, fractional parts, bounded gaps, sieve method, Bombieri–Vinogradov theorem.