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



increase in magnitude. In determining (by the formula li 2000 

 — li 1000) the number of primes between 1000 and 2000, 

 there would be an error in excess of about 2 ; so that if we 

 sum the actual primes between 1000 and 2000, in order to 

 obtain the number of primes between 1,000,000 and 4,000,000, 

 we might expect to obtain a number varying from the theo- 

 retical number of primes between these limits by from 2000 to 

 4000 ; but as it will be seen that the error is not so great, and 

 that it is not permanently in defect, we may infer that the for- 

 mula for computing the number of primes is more correct for 

 large numbers. 



The sum of the actual primes between 1000 and 2000 is 



205,054, which leaves, after deducting -(1999 — 997), the 



number 204,553 ; and the computed number of primes be- 

 tween 997^ or 994,009 and 1999^ or 3,996,001 is about 

 204,900. If we take in another prime, the difference lies in 

 the other direction; for the former number becomes 206,554, 

 and the latter about 205,908. 



To take an instance in which the number of primes has 

 been counted, we find the sum of the primes from 241 exclu- 

 sive to 503 inclusive (with the proper correction) to be 16,219, 

 and the number of primes from 241^ or 58,081 to 503^ or 

 253,009 to be 16,326 ; and taking the next prime, the numbers 

 are respectively 16,727 and 16,861. If we had begun one 

 prime earlier, the error would have lain in the other direction. 

 These examples sufficiently illustrate the general character of 

 the theorem. 



Case 2. Let (^x—x^\ then the sum of the squares of the 

 primes up to x inclusive is 



c^-\\{f)^\x^^ ^^log^-;^i(4(log^)2 + 2log «;) + ...; 



or the sum of the squares of the primes from y exclusive to x 

 inclusive is 



Xiif) -\\{f) + - {x^-y^) + - (a? log X -y log j/) nearly. 



Generally it will be found that the sum of the (wz — l)th pow- 

 ers of the primes from?/ exclusive to x inclusive is the number 



of the primes between^"' and x'^ increased by ^(^'"~^— y"~0» 



with further corrections involving the (m — 2)th and lower 

 powers of 7/ and x in combination with their logarithms, which 

 would be of small relative amount. 



By giving to m fractional values, we shall find that the sum 



