Weighted sums over almost-primes and the uniform Sathe-Selberg regime

For the weight J(n) to the power t, the sum over integers n up to x with Omega(n) = k has two forms. If k is fixed and a single exponent m at most k maximizes 1/a - t/a^2, the sum repeats Landau's formula at the abscissa (m-t)/m^2, with floor(k/m) blocks. If k runs over compacts inside 0 < k/log log x < m times 2 to the power (m-t)/m, the same sum has a uniform Sathe-Selberg expansion, depending on k modulo m. The case t = 0 recovers the classical barrier 2. At a transition point, two consecutive multiplicities coexist and the coefficient is given by a Gaussian saddle. The asymptotic symbol is used only where the amplitude is bounded away from zero. These results do not improve prime-gap estimates or the Goldbach conjecture. The record contains the English version (main file), a faithful Portuguese version, and the supplementary archive.Paper 3 of 3 on reciprocal exponents in the prime factorization. It depends on Paper 1.