Weighted equidistribution of factorial prime valuations in fixed power bands

Ma Yicen

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arXiv · 2609.37460

Weighted equidistribution of factorial prime valuations in fixed power bands

Ma Yicen

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Abstract

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!.

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Ma Yicen. 2026-09-27. Weighted equidistribution of factorial prime valuations in fixed power bands. https://arxiv.org/abs/2609.37460

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