Prime-Detecting Identities from Dirichlet Inversion

Abstract

For each integer $k\geq1$, let $σ_k(n)=\sum_{d\mid n}d^k$, let $σ_k^{-1}$ denote its Dirichlet inverse, and let $J_k$ denote the $k$th Jordan totient function. For $n\in\mathbb{N}$, define $F_k(n)=σ_k^{-1}(n)+μ(n)+J_k(n)+3$. We determine the zero set of $F_k$ completely. We prove that $F_1(n)=0$ if and only if $n$ is prime or $n=18$, whereas for every $k\geq2$, $F_k(n)=0$ if and only if $n$ is prime. The proof uses the convolution identities $σ_k=\mathbf{1}*I_k$ and $J_k=μ*I_k$, where $I_k(n)=n^k$, together with an explicit prime-power formula for $σ_k^{-1}$ and a classification by prime-exponent patterns. We also prove the companion characterization $σ_k^{-1}(n)+n^k=-1$ if and only if $n$ is prime, for $k\geq1$ and $n>1$. Lambert-series coefficient identities recover the three constituent functions, while an exact intermediate-divisor relation explains the connection between the two characterizations and the exceptional value $18$.