Explicit bounds for prime K-tuplets

TL;DR

This paper employs the large sieve method to derive explicit upper bounds on the count of prime k-tuplets up to a given number, enhancing understanding of their distribution.

Contribution

It provides the first explicit bounds for prime k-tuplets using the large sieve, advancing quantitative knowledge in prime number theory.

Findings

Derived explicit upper bounds for prime k-tuplets.

Applied large sieve technique to prime distribution problems.

Enhanced quantitative understanding of prime k-tuplets.

Abstract

Let $K\geq 2$ be a natural number and $a_i,b_i\in\mathbb{Z}$ for $i=1,\ldots,K-1$. We use the large sieve to derive explicit upper bounds for the number of prime $k$-tuplets, i.e., for the number of primes $p\leq x$ for which all $a_ip+b_i$ are also prime.