Multiplicative Functions on Shifted Primes

TL;DR

This paper investigates the behavior of positive multiplicative functions evaluated at shifted primes, showing under certain conditions that their values can be densely distributed in specific hypercubes, extending prior results in the field.

Contribution

It proves the density of shifted prime values of multiplicative functions in hypercubes, generalizing earlier work by De Koninck and Luca.

Findings

Values $f(p+1),\,\ldots,\,f(p+k)$ are dense in $[0,c]^k$ or $[c,+\infty)^k$ for some $c>0$.

The values $f(p+1),\ldots,f(p+k)$ can be ordered increasingly infinitely often.

The result applies to functions with prime values converging slowly to 1, where the sum of deviations diverges.

Abstract

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