ON PRIME NUMBEES. 351 



are there less than a given number x? More precisely, let 7r(a;) denote 

 the number of primes * not exceeding x : then tvhat is the order of ntagtii' 

 tilde of 7r(x) ? The Prime Number Theorem provides a complete answer 

 to this last question. It asserts that 



7r(cc) OO .- ' , 

 log X 



that is to say, that 7r(a;) and a; /(log x) are asymftotically equivalent, or 

 that their ratio tends to 1 when x tends to infinity. 



The first step towards the proof of this theorem was made by Euclid, 

 when he proved that the number of primes is infinite, or that 



7r(a;) -> oo. 

 Euclid's proof is classical, and can hardly be repeated too often. If 

 the number of primes is finite, let them be 2, 3, 5, . . ., P. The number 

 2. 3. 5. ... P + 1 is not divisible by any of 2, 3, 5, . . . ., P. It is 

 therefore prime itself, or divisible by some prime greater than P ; and 

 either alternative contradicts the hypothesis that P is the greatest prime. 

 It is worth remarking that Euclid's reasoning may be used to prove 

 rather more, viz. that the order of 7r(x) is at least as great as that of 

 log log a;.t 



The next advances were made by Euler, probably about 1740. It 

 was Euler to whom we owe the introduction into analysis of the Zeta- 

 function, the function on whose properties, as later research has shown, 

 the whole theory depends. 



Let s ^ (T + it. Then the function t,{s) is defined, when {r> 1, by the 

 equations 



C(s) = tn-' = 1-^- + 2-^- + 3'^ + . . . ; 



and Euler's fundamental contribution to the theory is the formula 



where the product extends over all prime values of f. Euler, it is true, 

 considered t,{s) as a function of a real variable only. But his formula 

 at once indicates the existence of a deep-lying connection between the 

 theory of t,{s) and the theory of primes. 



Euler deduced from his formula that the series %'P'\ obviously 

 convergent when s > 1, is divergent when s = 1 ; and from this it is 

 easy to deduce important consequences as to the order of ■it[x). It is 

 evident that 7r(x)<cc, so that the order of 7r(a;) certainly does not 

 exceed that of x, or, in the notation which is usual now, tt[x) = 0(a;).+ 

 It is an easy corollary of Euler's result that the order of 7r(x) is not 

 very much less than that of x ; that, for example, 7r(x) ;t 0{x'') for any 

 value of a less than 1 ; or again, more precisely, that 



^[x)izO{..-'^^\ 

 I (log x)^^" J 



(log x)i- 

 for any value of a greater than 1. 



* It proves most convenient not to count 1 as a prime. 

 t This was pointed out to me by Prof. H. Bohr of Copenhagen, 

 j f- 0{<p) means that the absolute value of/ is less than a constant multiple of <p 

 thus sin X = 0(1), 100 x -.^ 0(x). 



