[HM] Mertens Theorem

Stan Wagon (WAGON@macalester.edu)
Tue, 21 Sep 1999 12:57:56 -0500 (CDT)

I wish to share with you a most remarkable theorem from 1874 that I just ran
across. The famous Prime Number Theorem of 1896 says that the number of primes
less than x is asymptotic to 1/Log[x]. If one takes a probabilistic approach,
one is led, by the Sieve of Eratosthenes, to think that the number of primes
under x might be Product[1-1/p : p prime, p < Sqrt[x]]. This is because half
the numbers are discarded in the first pass, 1/3 of the remainder in the second
pass, then 1/5, and so on.

BUT: Mertens proved in 1874 that this product is asymptotic to 1.12/Log[x], the
"wrong" answer as far as the PNT goes. This elevates the PNT in my mind, since
it is by no means clear why the PNT's constant should be 1. The reason that the
probabilistic sieve overestimates the number of primes can be explained by the
fact that the true sieve of Eratosthenes is a MORE efficient sieve than one
would expect, because there is less overlap than expected. For example, let x =
10000. How many numbers are (simultaneous) multiples of 11, 23, and 47? NONE,
because these three multiply to more than x. But the probabilistic model would
assume some overlap for all such sets, and therefore it is less efficient in
its model of sieving.

Hardy and Wright observe that the Mertens Theorem is the reason that attempted
heuristic explanations of the PNT fail. The constant 1.12 above is in fact 2
E^(-EulerGamma).

Stan Wagon, Professor of Mathematics
Macalester College, St. Paul, Minnesota 55105
(651) 696-6057 fax = (651) 696-6518
primary e-mail: wagon@macalester.edu
secondary e-mail: wagon@compuserve.com
web page: http://www.math.macalester.edu/~wagon
or: http://141.140.2.74/~wagon