ON THE THEORY OF NUMBERS. 



363 



In the note of May 26, 1862, to which we have already referred, M. Kro- 

 necker has given other examples of the use of this method. Multiplying 

 together the three pairs of series, 



„!0??.=8S -?£— 82 "g eos2n * 

 W 2 i l-q Jn i l- 2 2 " ' 



co 

 H x cos ar=2i+ S [ji (2 "- lj2 + ji< 2 »+ 1 J 2 ]cos 2n#, 



^00^=82 M g" sip2w * 

 2 ! l + 2 2 " 



co 



H cosa;=2 (-l)"[ 2 i< 2 "+^ 2 - (Z i' 2 "-^ 2 jsin2n^ 



(i) 



(fi) 



2m-1 



2h+1 





1=42 R„2"" sin 2«.v, 

 9 1 



> 



(in) 



where R„=2 — ^+2~ * + q~^++. . . + q~i ( - 2n ~ 1)2 ,. 



/"TTT2TT 



and designating the definite integral lJ? f I _! cos # cfo by ti-I, we find 



9,1=4^1 r£^2±c/+»- 



1 L 2 H -2 _ 



1 



oo 



01=4^2 (-l)»nrr'tJZ2_, 

 1 <Z" + <T" 



00 2)1-1 2) t +l 



I = 42 E„r/' 2 f « * . + g 2 1 



Let r(») represent the sum of those divisors of n which do not surpass 

 V«, and which are even, or uneven, according as their conjugate divisors are 

 even or uneven ; and let f '(») represent the sum of the same divisors, each 

 divisor being taken positively or negatively according as the sum of itself and 

 its conjugate is unevenly or evenly even ; if » is a perfect square, we are to 

 replace y/nbyi^/n in the sums r(n) and r'(»)j we then obtain the ex- 

 pansions 



,«» co 



1 I=8 2 «S [>(») -r(n)] 2 », 0l=8r/2 T'(n)q n , 



1 1 



co n oo 



1 = 4^2,, S ft S,[2i[( 2 »- 1 )(2»-l + ^ + 4)-(2 M -l)2] +2 i[(2„ + ])(2n + l + 4 *)-(2 M -l)2n 

 110 ' 



co co 



= 4ryis $(4n)q"=8qil, F(n)q", 

 1 1 



