526 Mr. J. W. L. Glaisher on Applications of 



whence we find 



E,(»)-2(-l) 4CA - 1) A , -E 1 .(|) + 2(-l)« AB - 1) A'-B'-B r (^) 



_S(_i)«*w-»Aircn5 r ( 5 ^ c ) + . . . .=1, 



= or ( — V)^ n ~ l) n r , according as n is even or uneven. 



If r=0, E r (n) becomes E(t&), the excess of the number of 

 (4m + 1) divisors of n over the number of (4m + 3) divisors. 

 The function E(V) vanishes when n is of the form 4m + 3, 

 and is never negative. 



Formula involving E/(w), § 11. 

 § 11. Let e n =n r and f(x) = r— — - - v then 



X T X 



x 2 r x 2 S r x d 4^ 4 . 



F ^ = rT^ + i+? + i+^ + T+^ +&c - 



=w + E/ (2> 2 + E/ (3> 3 + E/ (4> 4 + &c. ; 



where E/ (n) denotes the excess of the sum of the rth powers 

 of those divisors of n whose conjugates are of the form 4m + 1 

 over the sum of the rth powers of those divisors whose con- 

 jugates are of the form 4m + 3. 

 Thus we have 



f(x)=£C — x 3 + X b — X 1 + &c., 



cj>(x) = x + 2 r x 2 + 3 r x* + 4V 1 + &c, 

 F {a) = x + E/ (2> 2 + E/ (3> 3 + E/ (4> 4 + &c., 



e n =n r , V'2n=0, 7] 2n+1 = ( — l) n ; 



whence we find 



E/ W -2a'E/ (J) +***< (J) -»«'E/ Qj) + . . . 



= or (— l) Kn-1) , according as w is even or uneven; 



E/( re )-2(-l)« A -'>E/(|)+2(-l)« AB -"E/(^) 



-j(-ir-«E/(^ c ) + ...=»'. 



It is easy to see that if n be uneven, 



E/« = (-l)^- 1) E^): 

 thus, when n is uneven, these formulas coincide with those of 

 the preceding section. 



y 



