6 Mr. C. W. Merrifield. Sums of the Series of the [Nov. 17, 



by giving x every prime value, and multiplying all the fractions on 

 one side of the equations into one another, and all the progressions on 

 the other side into one another, we have — 



. . . (primes) 

 =1+2+3+1+1+6+ . • • (natural numbers). 



For the multiplication of the progressions brings together once, and 

 once only, all the factors which make up each term of the harmonic 

 series. Moreover, this is true, not only of the simple numbers, but of 

 any powers, so that generally — 



( 1 _2-«)-i(l_3-«)-i(l_5-«)-i . . . (primes) 



= 1 + 2 _ " + 3" n + 4r n + . . . (natural numbers) . 



If n has any positive integral value except unity, the second side of 

 this equation is alwavs finite. When n=l 9 it may be transformed (as 

 is well known) into — 



i j .1 T , 1 



loo; e a: + 7 + — — + — .... 



& 1 2x 12b* 120a? 4 



•Calling the series of the reciprocal n th powers of the natural numbers 

 S n , and of the primes 2 W , we have, upon taking the logarithms, — 



2S»=2 w + -g-2.2 W + ij2g M + ;j2 4M -f- .... 



This would enable us to obtain the value of IS, if we knew the 

 series 2. The theorem of Mobius, mentioned above, enables us to 

 effect the reversal of this, and to obtain 2 in terms of IS. This 

 theorem, which is easily established inductively by indeterminate co- 

 efficients, is as follows : let — ■ 



=/(») + if <V>) + + if(x<) + 



then 



/(«0=F'(«) -PV) -PV) -+»(•») + + T VF<yo) _ 



the law of the last series being that every term whose index contains 

 a square factor disappears, the others being positive or negative ac- 

 cordingly as the index contains an even or odd number of prime 

 factors. Thus F(V), F(> 8 ), F(^ 9 ), F(> 12 ) all take the coefficient 

 zero, while ¥(x 6 ) and F(x 10 ) are positive ; but all the prime terms, 

 and such terms as F(« 30 ), F(a? 42 ), &c, are negative. This at once 

 gives — ■ 



2 tt =ZS« — jzlSyt — ^^3^ — ^S^-f- qISqh y^S^i "T'yV^lO" .... 



