Number of shifted primes as k-free integers

Abstract

Inspired by a result of Languasco on the number of representations of an integer as the sum of a prime number and a k-free integer under the absence of Siegel zeros, we study variants of a classical formula of Mirsky on the number of shifted prime numbers as k-free integers. Assuming that there are no Siegel zeros, our main result gives an asymptotic formula with a sharper error term for the number of prime numbers with two consecutive shifts that are simultaneously k-free. We improve the error term in this formula further by assuming weaker versions of the generalized Riemann hypothesis for Dirichlet L-functions. The flexibility of our approach also leads to unconditional formulas that are sensitive to the values at shifted prime numbers of a family of arithmetic functions with mild growth conditions. In the case of multiplicative functions, one obtains local to global type results on the value distribution of these functions at two consecutive shifts of prime numbers.