Weighted equidistribution of factorial prime valuations in fixed power bands
For a fixed modulus h and residue class c, we study the logarithmic mass of primes p in a fixed power band for which the exponent of p in n! is congruent to c modulo h. We establish the asymptotic ((b-a)/h + o(1)) log(n!) for primes satisfying n^a < p <= n^b, where 0 <= a < b <= 1. The argument combines a fixed-order derivative estimate, Vaughan's identity, and finite Fourier approximation of a discontinuous valuation function. We also derive a necessary capacity condition for infinite families of solutions to polynomial-factorial equations P(x) = n!.
math.NT↗