52 Analytical Researches coticerning Numbers, 



Case 4. Let (p^= -n; then the sum of the inverse squares 



of the primes up to x inclusive is 

 /» dx 



or 



dx , 1 _ 1 ^^S ^ _ 

 logo:' Ix^ 6 x^ 



The value of c is the sum of the series adinfinitunii for all the 

 other terms vanish when x is infinite. It was computed by 

 Euler at •454-2247. 



Generally, we shall find that the sum of the inverse nih 

 powers of the primes up to x inclusive is nearly 



''"*"^*\^^)+2^' 



where c is the sum ad iTifinitum ; and similar formulae apply 



to negative fractional powers. The logarithm-integrals may be 



found by means of Soldner's table and formulae. 



Case 5. Let <p^ = log^; then the sum of the logarithms of 



the primes up to x inclusive is 



/*, 1 , 1 \ogfx 

 c^Jdx+-\ogx+-~- .. 



or 



C + X+ ~ log X nearly. 



This theorem, which imports that the sum of the logarithms of 



the primes from 3^ exclusive to x inclusive is nearly x—i^ (more 



1 x\ 

 accurately x—y+ ;^log- ), may be verified by trial, and will 



w y / 



be found to be true of large numbers in an average sense. It 

 is true in the same sense as the formula for computing the 

 number of primes, and is in eff'ect identical with it. If we 

 take limits between which there are more than the average 

 number of primes, the sum of the logarithms will be too large, 

 and vice versa ; but as x increases, the sum of the logarithms 

 of the primes up to x becomes equal to x. 



Case 6. If <p,2?=log f 1 — \ , we shall find approximately 



