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Weighted equidistribution of factorial prime valuations in fixed power bands
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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BibTeX
@misc{yicen2026weighted,
title = {{Weighted equidistribution of factorial prime valuations in fixed power bands}},
author = {Ma Yicen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37460},
eprint = {2609.37460},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37460},
}
RIS
TY - UNPB
TI - Weighted equidistribution of factorial prime valuations in fixed power bands
AU - Ma Yicen
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.37460
UR - https://arxiv.org/abs/2609.37460
AB - 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!.
ER -