# Is there a “small $\omega$” number theorem?

In my studies of how primes jump (search this forum for a link), a question has been raised which may have been studied. Can anyone jump-start my literature search with references regarding the following problem?

Let $\omega$ denote the number of distinct prime factors of a number $n$, and consider the minimal value $w$ of this number as $n$ ranges over the interval (N,N+D), where D depends on N and might be an iterated logarithm of N or a fractional power of N.

Problem: for all N and a given small function D of N, what provable upper bounds do we have on $w$?

Of course, for infinitely many N and D a small constant, we have $w=1$. I look for results that apply for all N, but am willing to accept $w=2$ or some other small constant, or even $w=$ log log log N. Even a careful analysis of multiples of $\omega$-small numbers would be welcome.

• Cramer's conjecture gives you $\omega=1$ for D something like $C\log{(N\log{N})}^2$. Are you asking for unconditional results? – joro Oct 26 '16 at 11:18
• I am interested in conditional results, but I would also like explicit in D results. I might settle for w=C iterated log N for an unknown but provable constant C but I would want D to be 7N^{1/8} or better. Also, I am hoping to improve on known bounds between prime gaps; if I can't do that or improve upon bounds between products of two primes, maybe I can still get something on gaps between $\omega$-small numbers. Gerhard "Trying To Narrow Some Gaps" Paseman, 2016.10.26. – Gerhard Paseman Oct 26 '16 at 14:24
The integers in the interval $[n, n+D]$ have level of distribution close to $D$, hence you can apply a lower bound sieve (e.g. http://www.math.uiuc.edu/~ford/sieve_notes_intro_brun_hooley.pdf, Theorem BH.3) to show that this interval contains an integer, which has no small prime divisors. If $D=n^c$ with $c>0$, then you obtain some $\delta>0$ depending on $c$, such that $[n, n+D]$ contains an integer with smallest prime factor $>n^\delta$. In particular in this interval there is an integer with $\mathcal{O}(1)$ prime factors.