Multiplicative functions on shifted primes

Abstract

The understanding of the local behavior of arithmetic functions has been the subject of research of many mathematicians. Part of this research involves the study of the values of an arithmetic function on consecutive integers. For example, if we denote the divisor function by τ , in 1952 Erdös and Mirsky [1] asked whether the equation τ(n) = τ(n+1) admits infinitely many solutions in the set of natural numbers, a question that can be considered as a close relative of the twin prime conjecture. It remained open for about thirty years until Heath-Brown [3] answered it affirmatively in 1984 by showing that