Multiplicative functions on shifted primes

Abstract

Let f be a positive multiplicative function and let k≥2 be an integer. We prove that if the prime values f(p) converge to 1 sufficiently slowly as p→+∞, in the sense that ∑p|f(p)−1|=∞, there exists a real number c>0 such that the k-tuples (f(p+1),…,f(p+k)) are dense in the hypercube [0,c]k or in [c,+∞)k. In particular, the values f(p+1),…,f(p+k) can be put in any increasing order infinitely often. Our work generalises previous results of De Koninck and Luca.