episciences.org_8343_1653281509
1653281509
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
HardyRamanujan Journal
28047370
10.46298/journals/hrj
https://hrj.episciences.org
01
09
2022
Volume 44  Special...
A variant of the HardyRamanujan theorem
M. Ram
Murty
V
Kumar Murty
For each natural number $n$, we define $\omega^*(n)$ to be the number of primes $p$ such that $p1$ divides $n$. We show that in contrast to the HardyRamanujan 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 $x\to \infty$, $$\sum_{[p1,q1]\leq x} {1 \over [p1, q1]} \sim C \log x, $$ where the summation is over primes $p,q\leq x$ such that the least common multiple $[p1,q1]$ is less than or equal to $x$.
01
09
2022
8343
https://hal.archivesouvertes.fr/hal03251106v2
10.46298/hrj.2022.8343
https://hrj.episciences.org/8343

https://hrj.episciences.org/8343/pdf