Chap. X] SuM AND NUMBER OF DiVISORS. 297 
EM -s^.(^) +s^.(£) - . . . = (-i)^"-^>/v., 
E'Xn) -^a^E\(^^ +^a%^E',{^^ -... =^{-\r-'^'\ 
E\{n)-U-ir-'"'E\{^ +S(-l)(^«-i)/2^;(^JL^ - . . . =n^ 
where A,B,. . . are the odd prime factors of n. Note that ^ = or 1 according 
as n is even or odd. By means of these equations, each of the five functions 
(Tr{i^),- ■ -J E'r{n) is expressed in two or more ways as a determinant of 
order n. 
Ch. Hermite^° quoted five formulas obtained by L. Kronecker'^^ from the 
expansions of elHptic functions and involving as coefficients the functions 
$(n)=o-(n), the sum Z(n)_of the odd divisors of n, the excess ^(n) of the 
sum of the divisors >\/n of n over the sum of those <\/n, the excess 
$'(n) of the sum of the divisors of the form 8k=^l of n over the sum of the 
divisors of the form 8k^S, and the excess ^'(n) of the sum of the divisors 
8A;± 1 exceeding Vn and the divisors 8A;± 3 less than y/n over the sum of the 
divisors Sk=i=l less than \/n and the divisors 8A;± 3 exceeding \/n. Hermite 
found the expansions into series of the right-hand members of the five 
formulas, employing the notations 
Ei{x) = [a:+i] - [x], E^ix) = [x\[x+\]/2, 
a = l, 3, 5,...; 6 = 2,4,6,...; c = l, 2, 3,..., 
and A for a number of type a, etc. He obtained 
Z(l)+X(3)+. . .+X(A)=SE2(^), 
(r(l) +(7(2) + . . . + (7 (C) =SE2(C/c), 
^(l)+^(2)+. . .+^(0=2^2 (^'), 
X(2)+Z(4)+ . . . +Z(B) =is|a[^] +&^i[|]| 
«l>'(l)+*'(3)+. . .+$'(A)=S(-l)^"^-^)/«a[^], 
^'(l)+^'(3)+ . . . +^'{A) =S(-l)('^^+^>/«a| l^ ^+^^-^' j 
"BuU. Ac. Sc. St. Petersbourg, 29, 1884, 340-3; Acta Math., 5, 1884-5, 315-9. 
"Jour, fiir Math., 57, 1860, bottom p. 252 and top p. 253. 
