So I was given that if $$F(n)=\sum_{d|n}f(d)$$ then $$f(p^e)=F(p^e)-F(p^{e-1})$$ I'm trying to understand how to use this to actually solve specific problems. For example, let $F(n)=\sum_{d|n}f(d)$ and assume $F(n)$ is multiplicative, and $F(1)=1$. Find the formula for $f(n)$ when $F(p^e)=e+1$ for all primes p. By plugging this into the formula as best I can, I get $$f(p^e)=e+1-(e+1)=0$$ but this clearly isn't correct so I believe I'm making an error in the $F(p^{e-1})$ term, but I'm not sure how the change in exponent is supposed to affect it.
2026-02-22 20:34:12.1771792452
Inverse of a Multiplicative Arithmetic Function w/0 Mobius function
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You are making an error in the $F(p^{e-1})$ term. The rule is $F(p^n)=n+1$. Letting $n=e-1$, you get $F(p^{e-1})=(e-1)+1=e$. Therefore, $$ f(p^e)=F(p^e)-F(p^{e-1})=e+1-e = 1 $$