The Number Theoretic Omega Function and Summations Involving the Exponents of Prime Numbers in the Factorization of Factorials

The aim of this paper is to study the balancing of prime factors and their exponents in the standard factorization of n! into primes. We obtain explicit approximation for the sums ∑p⩽x(n)υp(n!)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{p\leqslant x(n)}\upsilon _p(n!)$$\end{document} and ∑p⩽x(n)υp(n!)logp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{p\leqslant x(n)}\upsilon _p(n!)\log p$$\end{document} for each boundary function x(n) with 2⩽x(n)⩽n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\leqslant x(n)\leqslant n$$\end{document}. Also, we estimate sums involving the exponents of prime factors in the factorization of factorials, and conclude that the sum of exponents of primes not exceeding elogn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {e}^{\sqrt{\log n}}$$\end{document} is asymptotic to the sum of the exponents of other primes in the factorization of n! into primes, as n→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty $$\end{document}. It is also shown that the product of primes not exceeding n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{n}$$\end{document} with their multiplicity is asymptotic to the product of other primes in the factorization of n! with their multiplicity, in logarithmic scale.