Erdős problem 448 · problems.science

Let $\tau(n)$ count the divisors of $n$ and $\tau^+(n)$ count the number of $k$ such that $n$ has a divisor in $[2^k, 2^{k+1})$ . Is it true that, for all $\epsilon > 0$ , $\tau^+(n) < \epsilon \cdot \tau(n)$ for almost all $n$ ?

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8 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

448.lean

Retained formal statement4 of 8

Hall and Tenenbaum [HaTe88] further prove that $\tau^+(n)/\tau(n)$ has a distribution function: there is a function F such that, for every $z$ , the set $\{n : \tau^+(n)/\tau(n) \le z\}$ has density F z.

FormalConjectures/ErdosProblems/448.leanErdos448.erdos_448.variants.hall_tenenbaum_distribution1 lineExact file

1∃ F, ∀ (z : ℝ), {n | ↑(Erdos448.tauPlus n) / ↑n.divisors.card ≤ z}.HasDensity (F z)

SolvedStatement only, no proof

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