M. Ram Murty ; V Kumar Murty - 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 - 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

  • 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 p1 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, nxω(n)2Cx(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, [p1,q1]x1[p1,q1]Clogx,
where the summation is over primes p,qx such that the least common multiple [p1,q1] is less than or equal to x.


Volume: Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021
Published on: January 9, 2022
Accepted on: January 9, 2022
Submitted on: August 7, 2021
Keywords: prime divisors,normal order,Brun’s sieve,Brun-Titchmarsh inequality,Primary 11N25, 11N36. Secondary 11N37, 11N64.,[MATH]Mathematics [math]
Funding:
    Source : OpenAIRE Graph
  • Funder: Natural Sciences and Engineering Research Council of Canada

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