On prime-producing sieves and distribution of $\alpha p-\beta$ mod $1$

Abstract

The author proves that there are infinitely many primes $p$ such that $\Vert \alpha p-\beta \Vert <p^{-\frac{28}{87}}$, where $\alpha$ is an irrational number and $\beta$ is a real number. This sharpens a result of Jia (2000) and provides a new triple $(\gamma ,\theta ,\nu )=(\frac{59}{87},\frac{28}{87},\frac{1}{29})$ that can produce special primes in Ford and Maynard's work on prime-producing sieves. Our minimum amount of Type-II information required ($\nu =\frac{1}{29}$) is less than any previous work on this topic using only traditional Type-I and Type-II information.

18 pages