10.46298/hrj.2022.8343
https://hrj.episciences.org/8343
Murty, M. Ram
M. Ram
Murty
Kumar Murty, V
V
Kumar Murty
A variant of the Hardy-Ramanujan theorem
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 $x$.
episciences.org
prime divisors
normal order
Brunâ€™s sieve
Brun-Titchmarsh inequality
Primary 11N25, 11N36. Secondary 11N37, 11N64.
[MATH]Mathematics [math]
2022-01-09
2022-01-09
2022-01-09
en
journal article
https://hal.archives-ouvertes.fr/hal-03251106v2
2804-7370
https://hrj.episciences.org/8343/pdf
VoR
application/pdf
Hardy-Ramanujan Journal
Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021
Researchers
Students