Additive uniqueness of $\mathtt{PRIMES}-1$ for multiplicative functions

TL;DR

This paper characterizes multiplicative functions satisfying specific additive relations on primes and their shifted sets, showing that such functions are either identity, constant, or zero except on certain squareful odd numbers.

Contribution

It provides a complete classification of multiplicative functions constrained by additive relations on primes and their shifted sets, revealing their structural form.

Findings

The only multiplicative functions satisfying the prime-based additive relation are identity, constant, or zero on certain numbers.

Functions satisfying the relation on primes minus one are necessarily the identity.

The paper establishes the uniqueness of these functions under the given conditions.

Abstract

Let $\mathtt{PRIMES}$ be the set of all primes. We show that a multiplicative function which satisfies \[ f(p+q-2) = f(p) + f(q) - f(2) \text{ for }p,q \in \mathtt{PRIMES} \] is one of the following: \begin{enumerate} \item $f$ is the identity function \item $f$ is the constant function with $f(n)=1$ \item $f(n)=0$ for $n \ge2$ unless $n$ is odd and squareful. \end{enumerate} As a consequence, a multiplicative function which satisfies \[ f(a+b) = f(a) + f(b) \text{ for }a,b \in \mathtt{PRIMES}-1 \] is the identity function.