A new characteristic of the identity function
The following Theorem is proved. Let \(f: \mathbb{N}\to \mathbb{C}\) be a multiplicative function such that \(f(1)=1\) and \(f(p+m^2)=f(p)+f(m^2)\) for all primes \(p\) and integers \(m\geq 1\). Then \(f(n)=n\) for all integers \(n\geq 1\). The elementary but ingenious proof uses sieve results and explicit numerical bounds for various prime number functions. The result should be compared with that of \textit{C. Spiro} [J. Number Theory 42, 232-246 (1992; Zbl 0756.11027)], where the same conclusion is drawn from the assumption \(f(p+q)=f(p)+f(q)\) (\(p\), \(q\) prime).
scientific article; zbMATH DE number 1011057
scientific article