Problem 448

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 \tau(n)

for almost all $n$ ?

Source: erdosproblems.com/448

Number theory
Divisors

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

DISPROVED (LEAN). The site's label; Erdős and Tenenbaum showed in 1981 that the integers with $\tau^+(n)<\epsilon\tau(n)$ do not have density one for small $\epsilon$ , and the Lean is a third-party formalization of their disproof, not built here, as the claim page below records.

Problem 448

Status

Proof claims

Discussion