Prime Factors – Upper Bound on Minimum Number of Prime Factors in Short Intervals

Question

Suppose that $H = H(X)$ is some quantity growing with $X$. Are there any bounds on $$F(X, H) = \min_{X < n\le X + H} \omega(n)?$$
It isn't hard to obtain a lower bound $\max_{x\sim X} F(X, H)\gg \frac{\log X}{H\log\log X}$. Also, for $H\gg X^\delta$, I believe it follows from a standard lower bound sieve the bound $F(X, H)\ll \delta^{-1}$. Is anything better known in either direction?

Answer 1

There is some useful information in the paper P. Erdôs and I. Kátai. On the growth of some additive functions of small intervals. Acta Math. Hungar. 33 (1979), 345-359.

Let $$O_{k}(n)=\max _{j=1, \ldots, k} \omega(n+j), \quad o_{k}(n)=\min _{j=1, \ldots, k} \omega(n+j).$$ Then (see I. Kátai. Local growth of the number of the divisors of consecutive integers. Publ. Math. Debrecen 18 (1971), 171-175.) for every $\varepsilon>0$, apart from a set of $n$'s having zero density, the inequalities $$ O_{k}(n) \leqq(1+\varepsilon) \varrho\left(\frac{\log k}{\log \log n}\right) \log \log n, \quad o_{k}(n) \geqq(1-\varepsilon) \bar{\varrho}\left(\frac{\log k}{\log \log n}\right) \log \log n $$ hold for every $k=1,2, \ldots$ Here $\varrho(u)$ $(u \ge 0)$ is defined as the inverse function of $\psi(z)=z \log \frac{z}{e}+1$ defined in $z \ge 1$, and $\bar{\varrho}(n)$ $(n \ge 0)$ is the inverse function of the same $\psi(z)$ defined in $0<z \le 1$. In the same paper it was conjectured that $$ O_{k}(n) \geqq(1-\varepsilon) \varrho\left(\frac{\log k}{\log \log n}\right) \log \log n $$ and $$ o_{k}(n) \leqq(1+\varepsilon) \bar{\varrho}\left(\frac{\log k}{\log \log n}\right) \log \log n $$

They prove that for every $\varepsilon>0$ these irequalities hold for every $k \ge 1$, apart from a set of $n$'s having zero density.