The distribution of intermediate prime factors (Q6633901)

Let \(\omega(n)\) and \(\Omega(n)\) be the number theoretic omega functions, counting distinct and total prime factors of an integer \(n>1\), respectively. Assuming that the prime factorization of \(n\) is written as\N\[\Nn =q_1^{a_1}q_2^{a_2} \cdots q_{\omega(n)}^{a_{\omega(n)}} =p_1p_2 \cdots p_{\Omega(n)},\N\]\Nwith \(q_1< q_2 < \cdots <q_{\omega(n)}\) and \(p_1 \leq p_2 \leq \cdots \leq p_{\Omega(n)}\), the \(\alpha\)-positioned prime factor of \(n\) is define by\N\[\NP^{(\alpha)}(n) := p_{\lceil \alpha \Omega(n)\rceil}.\N\]\NAccordingly, \(P^{\left(\frac 12\right)}(n)\) denote the middle prime factor of \(n\), taking into account multiplicity. In the paper under review, the authors determine an asymptotic for \(M^{(\alpha)}_{p}(x)\), counting the number of integers up to \(x\) that have \(p\) as the \(\alpha\)-positioned prime factor. Letting \(\beta :=(\log_2 p)/(\log_2 x)\) with \(\log_k x\) to be the \(k\)-fold iterate of the natural logarithm, their result for the middle prime factor asserts that, as \(p\to\infty\),\N\[\NM^{\left( \frac 12 \right)}_p(x)= \begin{cases}\displaystyle{C_\beta \displaystyle{\frac{x}{p(\log x)^{1-2\sqrt{\beta(1-\beta)}}\sqrt{\log_2 x}}}}\left(1+O_\varepsilon\left(\sqrt{\frac{\log_3 x}{\log_2 x}}\right)\right),\\\N~\\\N\displaystyle{C\displaystyle{\frac{x}{p(\log x)^{\frac 12 -\frac 32 \beta}}}}\left(1+O_\varepsilon\left(\sqrt{\frac{\log_3 x}{\log_2 x}} + \frac{(\log_2 p)^{-1/2}}{(\log p)^{\varepsilon^2}}\right)\right), \end{cases}\N\]\Nrespectively for the ranges \(\frac 15 +\varepsilon < \beta <1-\varepsilon\) and \(\beta<\frac 15-\varepsilon\), where \(\varepsilon>0\) and the constants \(C_\beta\) and \(C\) given by\N\[\NC_\beta=\frac{\exp\left(\frac{\gamma(1-2\beta)}{\sqrt{\beta(1-\beta)}}\right)}{\Gamma\left(1+1/\beta'\right)}\frac{\sqrt{\beta}+\sqrt{1-\beta}}{2\sqrt{\pi}\beta^{1/4}(1-\beta)^{3/4}}\prod_{q \text{ prime}}\left(1-\frac{1}{q}\right)^{\beta'}\left(1-\frac{\beta'}{q}\right)^{-1},\N\]\Nwith \(\beta'=\sqrt{(1-\beta)/\beta}\), and\N\[\NC=\frac{3e^{\frac{3\gamma}{2}}}{4\sqrt{\pi}}\prod_{q> 2 \text{ prime}} \left(1+\frac{1}{q(q-2)}\right)\approxeq 1.523555.\N\]\NAlso, they show that with \(\varphi =(1+\sqrt{5})/2\) and \(\varphi' =1/\varphi\), the average value of the logarithm of the middle prime factor of the integers satisfies\N\[\N\frac 1x \sum_{n \le x} \log P^{\left(\frac 12 \right)}(n) = A(\log x)^{\varphi'} \left(1+O\left(\frac{(\log_3 x)^{3/2}}{\sqrt{\log_2 x}}\right)\right),\N\]\Nwhere\N\[\NA=\frac{e^{-\gamma}}{\Gamma(\varphi+1)}\frac{\varphi + 1}{\sqrt{5}}\prod_p\left(1-\frac{1}{p}\right)^{\varphi'}\left(1-\frac{\varphi'}{ p }\right)^{-1} \approxeq1.313314.\N\]