CHAPTER X. 
SUM AND NUMBER OF DIVISORS. 
The sum of the A;th powers of the divisors of n will be designated crk(n) 
Often (r(n) will be used for (7i(n),and T{n) for the number ao{n)oi the divisors 
of n; also, 
!r(7i)=T(l)+T(2)+...+r(n). 
The early papers in which occur the formulas for T{n) and a{n) were cited 
in Chapter II. 
L. Euler^'^'^ applied to the theory of partitions the formula 
(1) p{x)='n.{l-x'')=s^l-x-x'+z^+x^-x'^-.... 
fc=l 
Euler'' verified for n<300 that 
(2) a{n)=(T{n-l)+(7{n-2)-a{n-b)-(j{n-7)+(7in-12)+.. ., 
in which two successive plus signs alternate with two successive minus 
signs, while the differences of 1, 2, 5, 7, 12, . . . are 1, 3, 2, 5, 3, 7, . . ., the 
alternate ones being 1, 2, 3, 4, . . . and the others being the successive odd 
numbers. He stated that (2) can be derived from (1). 
Euler^ noted that the numbers subtracted from n in (2) are pentagonal 
numbers (3a;^— a:)/2 for positive and negative integers x, and that if a(n—n) 
occurs it is to be replaced by n. He was led to the law of the series s by 
multipljdng together the earlier factors of p{x), but had no proof at that 
time that p = s. Comparing the derivatives of the logarithms of p and s, 
he found for —xdp/{pdx) the two expressions equated in 
,„. « nx"" x+2x^-bx^-1x^+l2x^^+ . . . 
{o) 2j ^= 
n=l \—X S 
He verified for a few terms that the expansion of the left member is 
(4) I a;V(n). 
n=l 
Multiplying the latter by the series s and equating the product to the numer- 
ator of the right member of (3), he obtained (2) from the coefficients of x". 
Euler® proved (1) by induction. To prove (2), multiply the left member 
of (3) by —dx/x and integrate. He obtained log p{x) and hence log s, 
and then (3) by differentiation. 
^Letter to D. Bernoulli, Jan. 28, 1741, Corresp. Math. Phys. (ed. Fuss), II, 1843, 467. 
''Euler, Introductio in Analysin Infinitorum, 1748, I, ch. 16. 
^Novi Comm. Ac. Petrop., 3, 1750-1, 125; Comm. Arith., 1, 91. 
^Letter to Goldbach, Apr. 1, 1747, Corresp. Math. Phys. (ed. Fuss), I, 1843, 407. 
»Posth. paper of 1747, Comm. Arith., 2, 639; Opera postuma, 1, 1862, 76-84. Novi Comm. Ac. 
Petrop., 5, ad annos 1754-5, 59-74; Comm. Arith., 1, 146-154. 
"Letter to Goldbach, June 9, 1750, Corresp. Math. Phys. (ed. Fuss), I, 1843, 521-4. Novi 
Comm. Ac. Petrop., 5, 1754-5, 75-83; Acta Ac. Petrop., 41, 1780, 47, 56; Comm. Arith., 
1, 234-8; 2, 105. Cf. Bachmann, Die Analytische Zahlentheorie, 1894, 13-29. 
279 
