On Oct 19, 2016, at 5:56 PM, rcs@xmission.com wrote:
Paraphrasing Hardy & Wright, section 22.18, Thm 437:
The percentage of numbers less than X which have K+1 prime factors is asymptotic to
(loglog X)^K ------------------ K! log X
Here, repeated primes are counted as additional prime factors, so 12 = 2 * 2 * 3 has three prime factors.
Fine point: The asymptotic percentage is the same if you don't include the numbers with square factors.
K=0 is the regular Prime Number Theorem.
Memory aid: The sum over K is the expansion for e^loglogX, minus 1, divided by logX. Or, think of the "excess" count of prime factors K being Poisson-distributed with mean loglog X.
Fun fact: loglog autocorrects to dogleg. -Veit
For large X, the percentage is nearly the same for X and cX, so we can loosely read this as saying that the probability that X has K prime factors is the same expression. The relative likelihood of two prime factors versus prime is a factor of loglogX. A third prime factor multiplies the likelihood by another factor of loglogX / 2. And so on.
H & W devotes 2.5 pages to the proof, but I think it boils down to writing out the sum of 1, taken over the appropriate set of N <= X, and approximating with the appropriate integral based on PNT.
I assume H & W is online somewhere. I'm working from the hardcopy 4th edition (1960), price 42 shillings, net. (!) I paid about $15; my local bookstore ordered it for me.
Rich
----------- Quoting Don Reble <djr@nk.ca>:
How does the density of semiprimes vary with the size?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun