280 History of the Theory of Numbers. [Chap, x 
Material on (1) will be given in the chapter on partitions in Vol. II. 
J. H. Lambert/ by expanding the terms by simple division, obtained 
n = l 1—X 
in which the coefficient of x"* is T{n). Similarly, he obtained (4) from the left 
member of (3). 
E. Waring^ reproduced Euler's^ proof of (2). 
E. Waring^ employed the identity 
n {x''-l)=x''-x'-'-x'-^+x'-^+x'-''- -...=A, 
k=l 
the coefficient of x^'", for v^n, being ( — 1)^ if v={3z^^z)/2 and zero if v is 
not of that form. If m^n, the sum of the mth powers of the roots of 
A=0 is a{m). Thus (2) follows from Newton's identities between the 
coefficients and sums of powers of the roots. He deduced 
m(m-l) ,o^ , m{m-l)im-2) m{m-l){m-2){m-S) 
\o) I (t{2)-\ a{S) o-(4) 
, ?n(m-l)(m-2)(m-3) ( .^ 
+ ■■■+ ^ {o-(2)j^-...= c-ml, 
where c= =•= 1 or is the coefficient of x^~"* in series A. Let 
U{x''-l)=x''-x'''-'-x''-^+x''-''-\-x''-^- . . . =A', 
where p ranges over the primes 1, 2, 3, 5, . . ., n. If m^n, the sum of the 
mth powers of the roots of A' = equals the sum a'{m) of the prime divisors 
of m. Thus 
ff'(m) =(r'(w- 1) +o-'(m-2) -t7'(m-4) -o-'(w-8) +(r'(m- 10) +o-'(m- 11) 
-(T'{m-12)-a'{m-lQ)+. . .. 
We obtain (5) with a replaced by a', and c by the coefficient of ic^'"*" in series 
A'. Consider 
n {x^^-l)=x^-x^-^ -x^-2'+x^-^'+ ... =5, 
with coefficients as in series A. The sum of the (Zm)th powers of the roots 
of B = equals the sum (T^^\m) of those divisors of m which are multiples of I. 
Thus ^ 
(T'^'\m)=(T'^'^{m-l)W\m-2l)-a^'\m-U)- . . ., 
with the same laws as (2) . The sum of those divisors of m which are divisible 
'Anlage ziir Architectonic, oder Theorie des Ersten und des Einfachen in der phil. und math. 
Erkenntniss, Riga, 1771, 507. Quoted by Glaisher.'* 
^Meditationes Algebraicse, ed. 3, 1782, 345. 
•Phil. Trans. Roy. Soc. London, 78, 1788, 388-394. 
