352 REPORTS ON THE STATE OJF SCIENCE. — ^11)15. 



It is also easy to prove that the order of ■7r{x) is definitely less than 

 that of X, or that, as we should express it now, Tr{x) = o{x).* This 

 theorem, when read in conjunction with those which precede, is, I think, 

 enough to suggest the Prime Number Theorem as a very plausible 

 conjecture, or at any rate to suggest that the true order is that of 

 a;/(log x). The theorem was in fact conjectured first by Gauss (1793) 

 and by Legendre (1798) ; and it is in Legendre's Essai sur la theorie des 

 nomhres that the conjecture first appears in print. 



In this state the problem remained for fifty years, until the publication 

 (1849-1852) of the researches of the Russian mathematician Tschebyschef. 

 I have no time to speak of Tschebyschef 's work as fully as it deserves, but 

 his chief results, in so far as they bear directly on the problem now before 

 us, were as follows : — 



(1) Tschebyschef showed that the problem is simplified if we take 



as fundamental not the function ■!r{x) itself, but the closely 

 related function 



6{x) = V log p 



p^x 



(the sum of the logarithms of all primes not exceeding x). He 

 showed that the order of 6{x) is the same as that of 7r(a;) log x, 

 and that the Prime Number Theorem itself is equivalent to 

 the theorem that 



6{x) (\i X. 



(2) He showed that 6{x) is actually of order x, and ^{x) of order 



a;/(log x), in fact that positive constants A and B exist such that 



' ^ <7:-(x)<B, ^ 



log X log x' 



(3) He showed that ifO{x)/x tends to a limit, then that limit must be 

 unity. 



What Tschebyschef could not prove is that the limit does in fact 

 exist, and, as he failed to prove this, he failed to prove the Prime Number 

 Theorem. And about Tschebyschef's methods (interesting as they are), 

 I shall say nothing ; for later research has shown that it was the essential 

 inadequacy of his methods which was responsible for his failure, and 

 that the theorem lies deeper in analysis than any of the ideas on which 

 he relied. ■ _ 



The next great step was taken by Riemann in 1859, and it is in 

 Riemann's famous memoir Ueber die Anzahl der Priiiizahlen unter einer 

 gegebenen Grosse that we first find the ideas upon which the theory has 

 now been shown really to rest. Riemann did not prove the Prime 

 Number Theorem: it is remarkable, indeed, that he never mentions it. 

 His object was a different one, that of finding an explicit expression 

 for 7r(x), or rather for another closely associated function, as a sum of 

 an infinite series. But it was Riemann who first recognised that, if 

 we are to solve any of these problems, we must study the Zeta-function 

 as a function of the complex variable s=^(t + it, and in particular 

 study the distribution of its zeros. 



* f=o(<p) means that fj^ >0. Thus siu x = o{x). This theorem also was stated 

 by Euler, but without satisfactory proof. 



