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.

Gerhard "Trying To Jump To Conclusions" Paseman, 2016.10.25.

  • $\begingroup$ Not exactly what you're asking for, but there are papers about the number of distinct prime factors of the product of all the numbers in a short interval, e.g., Erdős and Turk, Products of integers in short intervals, Acta Arith 44 (1984) 147-174, and Turk, Multiplicative properties of integers in short intervals, Nederl. Akad. Wetensch. Indag. Math. 42 (1980) 429-436. $\endgroup$ – Gerry Myerson Oct 25 '16 at 22:02
  • $\begingroup$ Thanks. The Turk reference sounds new to me. I will check out both of them. Gerhard "Will Hop, Skip, And Jump" Paseman, 2016.10.25. $\endgroup$ – Gerhard Paseman Oct 25 '16 at 22:17
  • $\begingroup$ Cramer's conjecture gives you $\omega=1$ for D something like $C\log{(N\log{N})}^2$. Are you asking for unconditional results? $\endgroup$ – joro Oct 26 '16 at 11:18
  • $\begingroup$ 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. $\endgroup$ – Gerhard Paseman Oct 26 '16 at 14:24
  • $\begingroup$ There are explicit strengthening of Cramer's conjecture, though they are quite strong: en.wikipedia.org/wiki/… $\endgroup$ – joro Oct 26 '16 at 15:30

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.

There has been a lot of work on the maximal distance between primes or integes with at most 2 prime factors, I do not know whether good bounds, that is, bounds which are better than those obtained by plugging in parameters into general theorems, exist e.g. for differences between integers with 20 prime factors.


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