Applications of bombieri-vinogradov type theorems to power-free integers

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 <= x such that p + 2y, 1 <= y <= 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.