Cg{n), arid certain series of arithmetical functions 271 



9. The following results may be proved in substantially the 

 same way: — 



V H' iq) log g _ 

 ^ <f>{q) 



4> i'l) 



^here A = U \l - ^_ — ^r^f ' 



CT running through all odd primes* ; 



y ^^ = t ^^5^-^^^^ = -2^; 



l,3,*5,-<^(?) ' 1,3,5,... </>(?) 



S^-^— =0, 



where v (q) is the number of different prime factors of q ; 

 ^ (-l)-(^)log9 _^. 



1,375,... ? ' 1,3,5,... 2 



2''("' y^ (») _ ^ 2''W^(,,)logn ^ ^ 2''(^V(n)(log7i)^ ^ ^ . 



x2j:!lii(!iM^' = -24^1og2. 



* 4 = -66016 . is the constant which appears to play a fundamental part in 

 'Goldbach's Problem'. See two notes, by Shah and Wilson and by Hardy and 

 Littlewood, in vol. xix (1919) of the Proceedings of the Cambridge Philosophical 

 Society (pp. 238-244 and 245-254). 



