Zahlen Aus Primzahlen

In 1791 Gauss made the following assertions (collected works, Vol. 10, p.ll, Teubner, Leipzig 1917): Primzahlen unter a ( = 00 ) a la Zahlen aus zwei Factoren lla· a la (warsch.) aus 3 Factoren 1 (lla)2a –- 2 la et sic in info In more modern notation, let 1tk(X) denote the number of integers not exceeding x which are made up of k distinct prime factors, k = 1, 2, …. Then his assertions amount to the asymptotic estimate x (log log X)k-l ( ) 1tk X '" –"';"'-"–-"::–:-'-,- (x-..oo). log x (k-1)! The case k = 1, known as the Prime Number Theorem, was independently established by Hadamard and de la Vallee Poussin in 1896, just over a hundred years later.