Popular differences in primes along fractional powers

TL;DR

This paper proves that primes have infinitely many pairs with differences in the Piatetski-Shapiro sequence for any non-integer c > 2, using an expectation estimate involving the von Mangoldt function.

Contribution

It establishes the existence of infinitely many prime pairs with differences in fractional power sequences for non-integer c > 2, extending previous results.

Findings

Primes contain infinitely many pairs with differences in fractional power sequences.

The expectation of the product of von Mangoldt functions approximates 1 with a specific error term.

The result applies to any non-integer c > 2, broadening the understanding of prime gaps.

Abstract

We prove that $\mathop{\mathbb{E}}_{m \leq M} \mathop{\mathbb{E}}_{n \leq N} \Lambda(n) \Lambda\bigl(n + \lfloor m^c \rfloor\bigr) = 1 + \rm{O}(\log^{2 - Bc} N)$, where $c > 2$ is a non-integer, $B \geq 3/c$, and $M$ is of order $N^{1/c} \log^{-B} N$. As a combinatorial consequence, we obtain that the primes contain infinitely many pairs whose difference belongs to the Piatetski-Shapiro sequence $\bigl\{\lfloor m^c \rfloor \colon m \in \mathbb{N} \bigr\}$ for any non-integer $c > 2$.