DATE: Tuesday, September 8 (2009), at 3:00pm
PLACE: LIT 305
SPEAKER: Krishnaswami Alladi
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TITLE:
On the normal number of prime factors of the integers
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ABSTRACT:
Even though prime numbers have been investigated
since Greek antiquity, the first systematic study of the number of
prime factos of the integers was only in 1917 due to Hardy and
Ramanujan - recent compared to the long history of number theory!
We will discuss their path-breaking paper
G. H. Hardy and S. Ramanujan, "The normal number of prime
factors of a number n", Quart. J. Math. Oxford (1917), 76-92
in which they show that almost all integers n have about loglog n
prime factors. One needs only upper bounds for the number of
primes up to x (Chebychev's estimate) and similar upper bounds
for certain weighted sums over primes, to establish this fundamental
result.
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