Claims

../

1981_01_01_erdos_tenenbaum: The Erdős–Tenenbaum theorem that the integers with tau^+(n) at most alpha tau(n) have upper density at most c(eps) alpha^(1-eps), so the conjectured density-one statement fails; Alexeev's Lean formalization is linked.

1988_09_15_hall_tenenbaum: The Hall–Tenenbaum theorem that tau^+(n)/tau(n) has a limiting distribution function nu with nu(z) of order between z/sqrt(log(2/z)) and z log(2/z), so the integers with tau^+(n) < eps tau(n) do not have density one.

---