Theorem χ(n) = d(n)/ln(n): New primality criterion with complexity improved from O(√n) to O(log n)

This record presents a new mathematical theorem that establishes a direct relationship between the divisor function d(n) and primality. The theorem proves that for any integer n > 1: χ(n) = d(n)/ln(n) Where: d(n) is the divisor counting function ln(n) is the natural logarithm Primality Corollary: If n is prime, then χ(n) = 2/ln(n) exactly. This property enables more efficient primality tests. Main Innovation: While traditional primality test methods have complexity O(√n) (requiring divisor checks up to √n), this theorem allows approximations with complexity O(log n) using the π(x) function (prime counting function).