On a problem of Chen and Liu concerning the prime power factorization of \(n!\) (Q2839299)

For \(p\) prime let \(e_p(n!)\) denote the power of \(p\) in the prime factorization of \(n!\). The study of the distribution properties of \(e_p(n!)\) goes back to \textit{P. Erdös} and \textit{R. L. Graham} in [Old and new problems and results in combinatorial number theory. Genève: L'Enseignement Mathématique, Université de Genève (1980; Zbl 0434.10001)], and the investigation was continued by subsequent authors. The present authors consider \(e_p(n^h!)\) for \(h=2\), \(h>2\) and \(e_p(q!)\) for \(q\) prime, and they answer questions raised by \textit{Y.-G. Chen} and \textit{W. Liu} in [J. Number Theory 82, No. 1, 1--11 (2000; Zbl 0999.11015)]. Theorem 2.1 of their paper concerns deriving asymptotic formulae when \(m\geq 1\) for \(\#\{n< x: n\equiv a\pmod d\), \(e_p(n^2!)\equiv r\pmod m\}\) and \(\#\{n< x: q\) prime, \(q\equiv a\pmod d\), \(e_p(q!)= r\pmod m\}\) valid for all \(d\geq 1\) and all \(0\leq a<d\), \(0\leq r< m\). The proof is based on work by \textit{C. Mauduit} and \textit{J. Rivat} [Acta Math. 203, No. 1, 107--148 (2009; Zbl 1278.11076)] and also their joint work with \textit{B. Martin} [Acta Arith. 165, No. 1, 11--45 (2014; Zbl 1395.11023)]. Let \(s_b(n^2)\) denote the sum of the digits of \(n^2\) in base \(b\); then an auxiliary result is required for certain exponential sums involving \(s_b(n^2)\) which generalizes Theorem 1 in the paper cited above.NEWLINENEWLINE Let \(h\geq 2\), \(p\) be prime, \(m\), \(d\geq 1\), \(0\leq a< d\), \(0\leq r< m\); then in Theorem 2.2 the authors obtain a lower bound valid as \(x\to\infty\) for \(\#\{n< x: n\equiv a\text{\,mod\,}d\), \(e_p(n^h!)\equiv r\pmod m\}\), and also show that \(\{e_p(n^h!)\pmod m: 0\leq n<C\), \(n\equiv a\pmod d\}=\mathbb Z_m\) with \(C= C(p,h,d,m)\) an effectively computable constant. Here the proof uses ideas of the second author [Funct. Approximatio, Comment. Math. 47, No. 2, 233--239 (2012; Zbl 1315.11010)].