CHAPTER X.
SUM AND NUMBER OF DIVISORS.
The sum of the Mh powers of the divisors of n will be designated <rk(ri) Often o-(n) will be used for tr^n), and r(n) for the number or0(n)of the divisors of n; also,
The early papers in. which occur the formulas for r(n) and a(ri) were cited in Chapter II.
L. Euler1*2'3 applied to the theory of partitions the formula
(1)
&=i
Euler4 verified for n<300 that (2) cr(n)=(r(n-l)+cr(n-2)-<r
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).
Euler5 noted that the numbers subtracted from n in (2) are pentagonal numbers (3z2—:r)/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 5 by multiplying together the earlier factors of p(x), but had no proof at that tune that p = s. Comparing the derivatives of the logarithms of p and 8, he found for —xdp/(pdx) the two expressions equated in
, £ nxn
He verified for a few terms that the expansion of the left member is (4) S xV(n).
n»l
Multiplying the latter by the series s and equating the product to the numerator of the right member of (3), he obtained (2) from the coefficients of xn. Euler6 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.
2Euler, Introductio in Analysin Infinitorum, 1748, I, ch. 16.
3Novi 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; Comra. 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.