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The following question naturally came up when dealing with 4-rank of certain class groups. In this case I want to inductively deal with some Legendre symbols, and to do so I want my squarefree integers to be "decently" spaced in the sense below.

Is there an absolute constant $C > 0$ such that for all functions $f$ going to infinity and almost all squarefree integers $n = p_1 \cdot \ldots \cdot p_r$ we have $$ \prod_{i = 1}^k p_i < p_{k + 1}^C $$ for all $f(n) \leq k < r$? What if we ask the question instead for almost all $k$?

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    $\begingroup$ What is $p_{i+1}$ in the RHS? $\endgroup$
    – Seva
    Commented Feb 28, 2020 at 18:55
  • $\begingroup$ Should have been $p_{k+1}$! $\endgroup$
    – P. Koymans
    Commented Feb 28, 2020 at 19:18
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    $\begingroup$ And, I suppose, you need $k<r$ instead of $k\le r$? (What is $p_{k+1}$ if $k=r$?) $\endgroup$
    – Seva
    Commented Feb 28, 2020 at 19:22

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This is not true for the all $k<r$ problem. Consider random $n$ below $x$, and put $z=\log x$. How many prime factors would a random number have in $[z,z^e]$? This is approximately Poisson with parameter $\sum_{z <p \le z^e} 1/p \approx 1$. So with positive probability you would find numbers with as many prime factors from this interval as you care to specify, which means that no fixed value of $C$ would work.

Look in the work of Ford on the multiplication table problem. There are different ways of phrasing conditions of this type; for example recasting it in terms of the number of prime factors up to some point satisfying a ``barrier" condition. Perhaps one of those reformulations would be useful for you.

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