ON PRIME NUMBERS. 



363 



Riemann proved 



(1) that ^(s) is an analytic function of s, regular all over the plane 



except for a simple pole at the point 1 ; 



(2) that ^(s) satisfies the functional equation 



^(1 -s) = 2(27r)--cos isTT r {s)^{s) ; 



(3) that ^(s) has zeros at the points - 2, - 4, — 6 ... ., and 



no other zeros except possibly complex zeros whose real parts lie 

 hetiveen and 1 inclusive. 



To these propositions he added certain others of which he could 

 produce no satisfactory proof. In particular he asserted that there is 

 in fact an infinity of complex zeros, all naturally situated in the ' critical 

 strip' 0^<7-<l; an assertion now known to be correct. Finally he 

 asserted that it was ' sehr wahrscheinlich ' that all these zeros have 

 the real part ^ : the notorious ' Riemann hypothesis ', unsettled to this 

 day. 



We come now to the time when, a hundred years after the conjectures 

 of Gauss and Legendre, the theorem was finally proved. The waywas 

 opened by the work of Hadamard on integral transcendental functions. 

 In 1893 Hadamard proved that the complex zeros of Riemann actually 

 exist ; and in 1896 he and de la Vallee-Poussin proved independently 

 that none of them have the real part 1, and deduced a proof of the Prime 

 Number Theorem. 



It is not possible for me now to give an adequate account of the 

 intricate and difficult reasoning by which these theorems are established. 

 But the general ideas which underlie the proofs are, I think, such as 

 should be intelligible to any mathematician. 



In the first place Buler's formula shows that log t,{s) behaves, through- 

 out the half-plane cr>l, much like the series ^p''. But ^{s) has a 

 simple pole for s = l, and so the sum of the series "^p'^'' tends 

 logarithmically to -f-oo when S -> through positive values. Suppose 

 now that (if possible) ^(l-t-«i)=0. Then the real part of 

 logt{l + 8 + ti), and therefore the real part of the series ^p"^"'"'', 



tends, also logarithmically, to — og when 8^0. It follows that the 

 series 



^ p~^~", — 2 2-'^^"' cos {t log p) 



tend to -)-oo with equal rapidity when S>0. As the first series is a 

 series of positive terms, while the signs of the terms in the second series 

 change with a certain regularity, it is natural to suppose that our last 

 conclusion is impossible ; and this is in fact not particularly difficult to 

 prove. 



I come now to the proof of the Prime Number Theorem itself. If 

 Ave differentiate Buler's formula logarithmically, we obtain 



^'(«) ^ ^ /log p ,logp \ s* L^g P . 



as) "^K p^ p'^ ■ ■ ■ 7 Z V"" ' 



where p assumes all prime values, m and n all positive integral values, 

 1915. A A 



