354 REPORTS ON THE STATE OF SCIENCE. — 1915. 



and A(n) is equal to log p if n is of tlie form p'" and to zero otherwise. 

 Let ij^ix) = 2 A(n) 



Then ip(x) is, for our present purpose, equivalent to 6(x) : it is easy to 

 show that the difference between the two functions is of order \/x. We 

 have therefore to prove that i/'(a;) Co x. 



The series on the right-hand side of the equation (1) is what is called 

 a ' Dirichlet's series ' ; and the theory of such series resembles the more 

 familiar theory of Taylor's series in one very important respect. We can 

 express the coefficients by contour integrals in which the function represented 

 by the series appears under the sign of integration. In particular we can 

 show that 



(^> *W = -2M|'J|T*' 



where the path of integration is a line parallel to the imaginary axis 

 and passing to the right of the point s=l. 



The general idea of the proof is now easy enough to grasp. Every 

 element of the integral (2) is of order x', where ff> 1 : we can therefore 

 draw no direct conclusion as to the behaviour of t//(cc) when x is large. 

 But it is at once suggested that we should try to make use of Cauchy's 

 theorem. The subject of integration has a simple pole at the point 1, 

 corresponding to the pole of C(s) itself, and the residue at the pole is 

 precisely x ; and there are no other singularities on the line cr=:l, 

 since t,{s), as we have seen, has no poles or zeros on that line. Suppose 

 then that we can move the path of integration across to the left of the 

 line, introducing the appropriate correction due to the pole. Plainly 

 we shall then have an expression for (//(x) — a? in the form of an integral 

 in which every element is of order less than that of x. And if we can 

 show that the same is true of the integral itself, we shall have proved 

 that i^(a;)c<j^) ihsit is to say, we shall have proved the Prime Number 

 Theorem. It will be observed that, if t,{s) had zeros whose real part 

 is equal to 1, then the result would be definitely false, since there would 

 be additional residues of order x. It thus becomes clear why the 

 older attempts to prove the theorem, without using the theory of func- 

 tions of a complex variable, were unsuccessful. 



The arguments which I have advanced are not exact : I have merely 

 put forward a chain of reasoning which seems likely to lead to the desired 

 result. The achievement of Hadamard and de la Vallee-Poussin was 

 to replace these plausibilities by rigorous proofs. It might be difficult 

 for me to make clear to you how great this achievement was. Some 

 branches of pure mathematics have the pleasant characteristic that 

 what seems plausible at first sight is generally true. In this theory 

 anyone can make plausible conjectures, and they are almost always 

 false. Nothing short of absolute rigour counts ; and it is for this reason 

 that the Analytic Theory of Numbers, while hardly a subject for an 

 amateur, provides the finest possible discipline in accurate reasoning for 

 anyone who will make a real effort to understand its results. 



