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Full text of "History Of The Theory Of Numbers - I"

CHAP. X]                        SUM AND NuMBEK OF DlVISOKS.                               297
^n)-Z(-l)u-1}*^                                  - . . . -1,
EM -2«r(j) +2^(5) - ... - (-D^'V,,
JE',(n) -S(~1)
where ^4 , 5, ... are the odd prime factors of n. Note that z^ == 0 or 1 according as n is even or odd. By means of these equations, each of the five functions <rr(n),. . ., E'r(ri) is expressed in two or more ways as a determinant of order n.
Ch. Hermite70 quoted five formulas obtained by L. Kronecker71 from the expansions of elliptic functions and involving as coefficients the functions $(ri)=(r(n), the sum X(ri)__of the odd divisors of n, the excess >P(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 8fc=±= 1 of n over the sum of the divisors of the form 8fc=*=3, and the excess ^'(n) of the sum of the divisors 8k=*= 1 exceeding \/n and the divisors 8/c=±=3 less than \/n over the sum of the divisors 8k=*=l less than -\/n and the divisors 8^=*= 3 exceeding ^/n. Hermite found the expansions into series of the right-hand members of the five formulas, employing the notations
Ei(x) = [05+i] - W,          JEa(aO = [x][a?+l]/2,
0 = 1,3,5,...;          6 = 2,4,6,...;         c=l,2,3,...,
and A for a number of type a, etc.   He obtained . . . +X(A) = . + a (C) =
"Bull. Ac. So. St. P<§tersbourg, 29, 1884, 340-3; Acta Math., 5, 1884-5, 315-9. 71 Jour, fur Math., 57, 1860, bottom p. 252 and top p. 253.