M. Ram Murty ; V Kumar Murty
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A variant of the Hardy-Ramanujan theorem
hrj:8343 -
Hardy-Ramanujan Journal,
January 9, 2022,
Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021
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https://doi.org/10.46298/hrj.2022.8343
A variant of the Hardy-Ramanujan theoremArticle
Authors: M. Ram Murty 1,2; V Kumar Murty 3
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M. Ram Murty;V Kumar Murty
1 Department of Mathematics and Statistics, Queen’s University
2 Departement of Mathematics and Statistics [Kingston, Queen's University]
3 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 loglogn, the function ω∗(n) does not have a normal order. We conjecture that for some positive constant C, ∑n≤xω∗(n)2∼Cx(logx).
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]≤x1[p−1,q−1]∼Clogx,
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.