{"docId":8343,"paperId":8343,"url":"https:\/\/hrj.episciences.org\/8343","doi":"10.46298\/hrj.2022.8343","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":607,"name":"Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-03251106","repositoryVersion":2,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-03251106v2","dateSubmitted":"2021-08-07 18:16:33","dateAccepted":"2022-01-09 17:45:33","datePublished":"2022-01-09 17:48:25","titles":{"en":"A variant of the Hardy-Ramanujan theorem"},"authors":["Murty, M. Ram","Kumar Murty, V"],"abstracts":{"en":"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 $x$."},"keywords":[{"en":"prime divisors"},{"en":"normal order"},{"en":"Brun\u2019s sieve"},{"en":"Brun-Titchmarsh inequality"},"Primary 11N25, 11N36. Secondary 11N37, 11N64.","[MATH]Mathematics [math]"]}