Erdős 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 \(\epsilon > 0\), \[ \tau^+(n) < \epsilon \cdot \tau(n) \] for almost all \(n\)? This is false, and was disproved by Erdős and Tenenbaum [ErTe81], who showed that in fact the upper density of the set of such \(n\) is \(\asymp \epsilon^{1-o(1)}\) (where the \(o(1)\) in the exponent \(\to 0\) as \(\epsilon \to 0\)). A more precise result was proved by Hall and Tenenbaum [HaTe88] (see Section 4.6), who showed that the upper density is \(\ll \epsilon \log(2/\epsilon)\). Hall and Tenenbaum further prove that \(\tau^+(n)/\tau(n)\) has a distribution function. Erdős and Graham also asked whether there is a good inequality known for \(\sum_{n \leq x} \tau^+(n)\). This was provided by Ford [Fo08] who proved \[ \sum_{n \leq x} \tau^+(n) \asymp x\frac{(\log x)^{1-\alpha}}{(\log\log x)^{3/2}} \] where \[ \alpha = 1-\frac{1+\log\log 2}{\log 2} = 0.08607\cdots. \]