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Abstract

Studying a variant of a classical result of Walfisz on the number of representations of an integer as the sum of a prime number and a square-free integer with an extra additive constraint on the prime summand, we obtain an asymptotic formula for the number of representations of an integer $N$ such that $N-1$ is a prime number in the form $p+N-p$, where $p$ is a prime number, $N-p$ is square-free and $p-1$ is cube-free. We improve the error term for the number of representations of an integer as the sum of a prime number and a $k$-free integer conditionally by assuming weaker forms of the Riemann hypothesis for Dirichlet $L$-functions. As a further application of our method, we find an asymptotic formula for the number of prime numbers $p\leq x$ such that $p+2j$, $1\leq j\leq 7$, are all square-free. Our formula shows that a positive proportion of prime numbers leads to a longest possible progression of eight consecutive odd, square-free integers. A key ingredient in our approach is the Bombieri–Vinogradov theorem and its variant for sparse moduli due to Baier and Zhao which regulates the uniform distribution of prime numbers along certain short arithmetic progressions.