Prime-Detecting Identities from Dirichlet Inversion
Abstract
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.
Related
- On the trace identities of 2-dimensional tr-prime algebras over a finite field
- The variance of the Pitman--Yor process: Cifarelli--Regazzini-type identities and inversion formulas
- Large values of cubic and quartic Dirichlet L-functions of prime moduli
- Extreme values of Dirichlet L-functions on the critical line with a growing prime modulus
- The Algebra of Parity-Twisted Crank Moments and a Prime-Detecting Expression
BibTeX
@misc{liu2026primedetecting,
title = {{Prime-Detecting Identities from Dirichlet Inversion}},
author = {Zhichen Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33278},
eprint = {2609.33278},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33278},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Prime-Detecting Identities from Dirichlet Inversion
AU - Zhichen Liu
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.33278
UR - https://arxiv.org/abs/2609.33278
AB - 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.
ER -