The distribution of intermediate prime factors

Abstract

Let $P^{\left(\frac{1}{2}\right)}(n)$ denote the middle prime factor of $n$ (taking into account multiplicity). More generally, one can consider, for any $\hat{I}\pm \in (0,1)$, the $\hat{I}\pm$-positioned prime factor of $n$, $P^{(\hat{I}\pm )}(n)$. It has previously been shown that $\log \log P^{(\hat{I}\pm )}(n)$ has normal order $\hat{I}\pm \log \log x$, and its values follow a Gaussian distribution around this value. We extend this work by obtaining an asymptotic formula for the count of $n\le x$ for which $P^{(\hat{I}\pm )}(n)=p$, for primes $p$ in a wide range up to $x$. We give several applications of these results, including an exploration of the geometric mean of the middle prime factors, for which we find that $\frac{1}{x}{\sum }_{1<n\le x}\log P^{\left(\frac{1}{2}\right)}(n)\sim A(\log x{)}^{\ddot{I}†-1}$, where $\ddot{I}†$ is the golden ratio, and $A$ is an explicit constant. Along the way, we obtain an extension of Lichtman's recent work on the ``dissected'' Mertens' theorem sums ${\sum }_{\begin{array}{c}{P}^{+}(n)\le y\\ \hat{I}\text{c◯}(n)=k\end{array}}\frac{1}{n}$ for large values of $k$.