A variant of the Hardy-Ramanujan theorem

Abstract

For each natural number $n$, we define $\omega^*(n)$ to be the number of primes $p$ such that $p-1$ divides $n$. We show that in contrast to the Hardy-Ramanujan theorem which asserts that the number $\omega(n)$ of prime divisors of $n$ has a normal order $\log\log n$, the function $\omega^*(n)$ does not have a normal order. We conjecture that for some positive constant $C$, $$\sum_{n\leq x} \omega^*(n)^2 \sim Cx(\log x). $$ Another conjecture related to this function emerges, which seems to be of independent interest. More precisely, we conjecture that for some constant $C>0$, as $x\to \infty$, $$\sum_{[p-1,q-1]\leq x} {1 \over [p-1, q-1]} \sim C \log x, $$ where the summation is over primes $p,q\leq x$ such that the least common multiple $[p-1,q-1]$ is less than or equal to $x$.