Mr. C. J. Hargrcave on the Law of Prime Numbers. 117 

 \og{\p) = -hs^l-^^')-hs(l-~)-log(l-^-... 



21+P ^ 31+P ^ 51+p ^ 71+p ^ 

 + certain quantities which have a finite limit. 

 We may therefore conclude that the series 



21+P 31 +p 51+P 7'+p 



differs from the logarithm of p-^ by a quantity, which, as p di- 

 minishes without limit, approximates to a certain finite constant. 

 It will also be readily perceived, by integrating the function 

 Xp with respect to p, that the expression 



111 



+ TT—TT^TTTT-. + ^T—ii^rTT:; + •••; or 



--/\pdp + p, 



(log2)2^+p ' (log 3)3' +p ' (log4)4i+p 



approximates to —logp-{-{l—ry)p+ a constant: from which 

 we infer that 



{log n)n^+p* 

 where n has every value from 2 to oc , differs only by a constant 

 from S — fip, where fi has the value of every prime number from 



2 upwards, and p is supposed to be as small as we please ; so 

 that, when p = 0, these two expressions, which are infinite and 

 of the order —• log p, differ only by a finite constant. Or 



fjL n log n 



But S -T is in reality the sum of the reciprocals of a series 



n log n '' 



of numbers, each of which is equal to the one before it increased 



by its logarithm (see Phil. Mag. vol. xxxv. p. 49) ; and we thus 



see that the sum of the reciprocals of the prime numbers differs 



only by a constant from the sum of the reciprocals of a series of 



numbers which follow the law P„+i = P^ + logP„, the magnitude 



of the constant being dependent on the point at which the latter 



series is made to commence. If we call the series which obeys 



the law P^+ logP^ = l\+i the series of theoretical primes, and 



assign a proper value to the first term of the series, we may then 



assert that the infinite sum of the reciprocals of the real primes 



is equal to the infinite sum of the reciprocals of the theoretical 



primes ; and we may, in the sense before indicated, assert that 



the average distance between two primes at the point x in the 



ordinal series is log x. 



