Cg{n), and certain series of arithmetical functions 269 

 or zero, according as n is or is not of the form p*' ; or 



which contradicts (7-1). 



It should be noticed that, if we are merely concerned to prove 

 that (7-1) is false, all the difficult part of the preceding argument 

 may be omitted. For, if (7-1 ) were true,/(s) would certainly tend 

 to the limit A {n) when s -^ 1, and the mere knowledge of the value 

 of /(I) is enough to show that this is not the case. 



The simplest examples of the formulae (6-1) and (7-3) are 

 obtained by taking ?i = 2, when 



is equal to ^Ji{q)'\i q is odd and to - fM{q) if q is oddly even. We 

 then obtain 



V (-1)^~M/^(9)J ^ (1 _ 2i-«) ^(s) = l-« - 2-« + 3-« - ..., 



(f>s iq) 



which tends to the limit log 2 when s -* 1 ; but 



shi)r^(2)i=ii„g2. 



The last result may be written in the form 



1 1 _1 ]_ _1 1_ ^1, 2 



^-f(2) + ^T3)'^«^{5) <^(6)^</>(7) <^(10) ••• ^ ^^ • 



The series ^^. 



8. Another very interesting series, the consideration of which 

 is suggested naturally by the work which precedes, is the series 



T(q)~ ^(2) 0(3) </)(5) </)(6) 

 I shall prove that this series is convergent, and has the sum : 

 more generally, 



2^ = (8-1). 



Let f{s) = ^ "^ ■ 



q'-' </> iq) ' 



and let us write s — l=z. Then 



f{s) = Ux^, 



where X^= 1 + ^^^|^) + ^^^ <^ (t!7^) ^ •- " 



* We agree to regard ij.(x), when x is not an integer, as zero. 



18—2 



