A VARIANT OF THE HARDY-RAMANUJAN THEOREM

M. Ram Murty (1) , V Kumar Murty (2)

1 Department of Mathematics and Statistics, Queen’s University
2 Department of Mathematics [University of Toronto]

For each natural number n, we define ω * (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 ω(n) of prime divisors of n has a normal order log log n, the function ω * (n) does not have a normal order. We conjecture that for some positive constant C, n≤x ω * (n) 2 ∼ 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 → ∞, [p−1,q−1]≤x 1 [p − 1, q − 1] ∼ C log x, where the summation is over primes p, q ≤ x such that the least common multiple [p − 1, q − 1] is less than or equal to x.

Domaines: Mathématiques [math]

M. Ram Murty, V Kumar Murty. A VARIANT OF THE HARDY-RAMANUJAN THEOREM. 2021. ⟨hal-03251106v1⟩