Re: Maximal number of distinct factors

From: Robin Chapman (rjc_at_ivorynospamtower.freeserve.co.uk)
Date: 11/23/04 

Date: Tue, 23 Nov 2004 16:30:26 +0000

Abey wrote:

>
> Given a number n, is there a tight upper bound on the number of its
> distinct factors (need NOT be prime), d(n)?
>
> Specifically, is d(n) = O(logn)?

No

> ps: I offer my apologies if this happens to be a repost.

It is, and so I post here my recent s.m.r. reply (hasn't yet
turned up there )
See Hardy & Wright. In the 5-th edition:

Theorem 314 states that d(n) is not O(log(n)^k) for any k.

Theorem 315 states that d(n) is O(n^k) for any k > 0.

Theorem 317 states that
limsup (log d(n) log log n/log n) = log 2,
and so for k > 0
d(n) < 2^{(1+k)log n/log log n}
for all large n, and
d(n) > 2^{(1-k)log n/log log n}
for infinitely many n. 

-- 
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9"
Francis Wheen, _How Mumbo-Jumbo Conquered the World_