Multiplicative Functions on Shifted Primes

Let $f$ be a positive multiplicative function and let $k\ge 2$ be an integer. We prove that if the prime values $f(p)$ converge to $1$ sufficiently slowly as $p\to +\mathrm{\infty }$, in the sense that ${\sum }_p\mathrm{\mid }f(p)-1\mathrm{\mid }=\mathrm{\infty }$, there exists a real number $c>0$ such that the $k$-tuples $(f(p+1),\dots ,f(p+k))$ are dense in the hypercube $[0,c{]}^k$ or in $[c,+\mathrm{\infty }{)}^k$. In particular, the values $f(p+1),\dots ,f(p+k)$ can be put in any increasing order infinitely often. Our work generalises previous results of De Koninck and Luca.