Critical Prime-Index Support-Span Laws for Integers with a Fixed Number of Distinct Prime Factors

Abstract

Let N>=2 range over the integers as the size cutoff, and fix an integer r>=2. For an integer n having exactly r distinct prime factors, we measure the number K of omitted global prime indices between its smallest and largest support primes. With M_r(N)=pi(N^(1/r))-pi(N^(1/(r+1))), when K/M_r(N)->tau in [0,infinity), the counting problem has a nontrivial phase function A_r(tau). The manuscript derives the sequential finite-tau law and its compact-uniform strengthening over every compact subset of [0,infinity), with the same compact-uniform law for exact-to-relaxed fidelity and its defect complement. The parameter r is fixed throughout; K is a nonnegative integer; all N-dependent asymptotic assertions are eventual in N; no assertion is made at tau=infinity or uniformly in varying r.