来自Dirichlet逆的素数检测恒等式
Prime-Detecting Identities from Dirichlet Inversion
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中文总结 AI 辅助
本文通过Dirichlet逆与Jordan totient函数构造恒等式，完全刻画其零点集，证明$F_1(n)=0$当且仅当$n$为素数或$n=18$，且$k\geq2$时$F_k(n)=0$当且仅当$n$为素数，并给出伴随刻画。
中文摘要 AI 辅助
对于每个整数$k\geq1$，设$\sigma_k(n)=\sum_{d\mid n}d^k$，设$\sigma_k^{-1}$表示其Dirichlet逆，并设$J_k$表示第$k$个Jordan totient函数。对于$n\in\mathbb{N}$，定义$F_k(n)=\sigma_k^{-1}(n)+\mu(n)+J_k(n)+3$。我们完全确定了$F_k$的零点集。我们证明$F_1(n)=0$当且仅当$n$是素数或$n=18$，而对于每个$k\geq2$，$F_k(n)=0$当且仅当$n$是素数。证明使用了卷积恒等式$\sigma_k=\mathbf{1}*I_k$和$J_k=\mu*I_k$，其中$I_k(n)=n^k$，以及$\sigma_k^{-1}$的显式素数幂公式和按素数指数模式的分类。我们还证明了伴随刻画$\sigma_k^{-1}(n)+n^k=-1$当且仅当$n$是素数，对于$k\geq1$且$n>1$。Lambert级数系数恒等式恢复了三个组成函数，而一个精确的中间除数关系解释了这两个刻画与例外值$18$之间的联系。
英文摘要
For each integer k >= 1, let sigma_k(n) = sum_{d|n} d^k, let sigma_k^{-1} denote its Dirichlet inverse, and let J_k denote the kth Jordan totient function. Using established prime-power values of sigma_k^{-1}, we first obtain the elementary criterion sigma_k^{-1}(n) + J_k(n) + 2 = 0 precisely when n is prime. At k = 1, this involves Euler's totient J_1 = phi and provides a prime-only comparison for the next equation. Our main result classifies the zeros of sigma_k^{-1}(n) + mu(n) + J_k(n) + 3: for k = 1, they are precisely the primes and 18; for k >= 2, they are precisely the primes. The proof separates integers according to their prime-exponent patterns. Sign and divisibility arguments exclude composite zeros outside the cube-free nonsquarefree case, where an additional divisibility condition isolates 18 at k = 1 and precludes composite zeros for higher k. Standard convolution and Lambert-series identities provide the arithmetic framework, while an intermediate-divisor relation explains the exceptional zero. These criteria are structural characterizations rather than efficient primality tests, since direct evaluation of the multiplicative formulas generally presupposes a factorization of n.
