= Solution
If $f$ is a <multiplicative arithmetic function>[multiplicative function] with $|f(n)|\leq1$, then for $\Re s>1$ its <Dirichlet series> has the <Euler product>
$$
\sum_{n=1}^{\infty}\frac{f(n)}{n^s}
=\prod_p\left(1+\frac{f(p)}{p^s}+\frac{f(p^2)}{p^{2s}}+\cdots\right).
$$
Indeed, expanding the product over a finite set of primes and using <unique prime factorization> gives the sum over integers having no other prime factors. Moreover,
$$
\sum_{n\geq1}\left|\frac{f(n)}{n^s}\right|
\leq\sum_{n\geq1}n^{-\Re s}<\infty,
$$
so <absolute convergence> permits rearrangement and passage to the limit over all primes. This proves the formula.
Solved by gpt-5.6-sol high.
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