erdos_457.variants.one_sub

Taking $n$ to be the product of primes between $\log n$ and $(2 + o(1)) \log n$ gives an example where $$ q(n, \log n) \ge (2 + o(1)) \log n. $$ Can one prove that $q(n, \log n) < (1 - \epsilon) (\log n)^2$ for all large $n$ and some $\epsilon > 0$?

theorem erdos_457.variants.one_sub : answer(sorry) ↔ ∃ ε > (0 : ℝ),
    ∀ᶠ n in Filter.atTop, q n (Real.log n) < (1 - ε) * Real.log n ^ 2

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