118 Mr. C. J. Hargreave on the Law of Prime Numbers. 



A practical Method of ascertaining the exact number of Primes up 

 to an advanced point in the Ordinal Series. 



In the paper above referred to, I verified the formula \ix by 

 computing it for various values of a? up to a million ; and com- 

 pared the results with those derived from actually counting the 

 primes up to that point from Burckhardt's and Chernac's tables. 

 The number of primes up to a million, as counted, appeared to 

 be 78,493, the formula giving 78,626. Since that period, I 

 ascertained, by counting, the number of primes between two 

 millions and three millions to be 67,751, the formula giving 

 67,916. 



I now propose to point out a method of ascertaining the num- 

 ber of primes up to a point considerably beyond the limits of 

 any tables ; and to apply it to the number 10,000,000 and in- 

 ferior numbers. 



Let us denote by Pa? the exact number of primes inferior to x, 



(X\ sc 



— j the number — when integer, and the next whole 



number below - when it is not integer. If we take at random 



P 

 a set of prime numbers jo,, jSg, jOg . , .pm the number of numbers 



between and x which are not divisible by any of the set /?„ 



p^ . . .pny is 



.-x{»(^^)}..{»r|)}-x{»(i)}.... 



±n( ^— ); 



the 2 being intended to cover every combination of one, of two, 

 of three, &c., as the case m^y be, taken'out of the setpi,j9j,.,j9„. 

 For example, the number of numbers up to 1 0,000 not contain- 

 ing either 2, 3, 5, 7 or 11, will be 



10,000 



- (5000 + 3333 + 2000 -f 1428 + 909) 



+ (1666+1000+714+454+666+476+303+285 + 181 + 129) 



-(333 + 238 + 151 + 142 + 90 + 64+95 + 60 + 43 + 25) 



+ (47 + 30 + 21 + 12 + 8) 



-4. 



This proposition is obvious from the considerations developed 

 at pages 38 and 39 of the former paper, and its truth will im- 

 mediately suggest itself on consideration*. 



* The theorem may be generalized as follows. Denote by An the sum 



