48 Mr. C. J. Hargreave's Analytical Researches 



us a perfect idea of the rate at which primes occur. Thus if 

 we wish to know at what point of the ordinal series the primes 



1000 



come at the rate of 50 per thousand, the answer is e*" 

 or e^°, which is about 485,000,000 ; that is, for many millions 

 before and after 485 millions, the rate of primes is 50 to the 

 thousand. 



For ranges of moderate magnitude, the formula 



af{\ogx' — \)—x{\ogx — l) 

 will be found to represent the number of primes between a?' 

 and X with considerable accuracy in any part of the series. 

 For since log x is the average distance between two primes 

 at Xi the average of these average distances from x to x' will 

 be 



a^([ogx^—\)—x(\ogx—l) 

 a}—x ' 



and xl x^^x be divided by this expression, we obtain the for- 

 mula above written as the average number of primes between 

 y and X. This would give us for the primes between x and 

 Ix the expression 



^ ^ X 



log^ + 2log2 — 1 *^^ log a?+ -38629436' 

 which will be found to be nearly correct. The above formula 



for any number up to about 1200,000,000 by one application of the for- 

 mula for li*' — \\x. 



