History Of The Theory Of Numbers - I

CHAP. X] SUM AND NUMBER OP DlVISOBS. 281
by the relatively prime numbers a, b, e, ... is
(m) ~S<r(^ (m) +Sa(ab<:) (m) - ....
Waring noted that <r(a/?) =a<rQ3) -f(sum of those divisors of 0 which are not divisible by a). Similarly,
<r(a/3y. . .) =aff(fty. . .) -f(sum of divisors of fry. . . not divisible by a) = a/3cr(7d . . .) + (sum of divisors of ]8y . . . not divisible by a) +a(sum of divisors of 7$ ... not divisible by 0),
etc. Again, cr(I)(a/3) = a<ra)03) + (sum of divisors of /3 divisible by I but not by a). The generalization is similar to that just given for a. C. G. J. Jacobi10 proved for the series s hi (1) that
Jacobi11 considered the excess E(n) of the number of divisors of the form 4ra+l of n over the number of divisors of the form 4ra+3 of n. If n=2puv, where each prime factor of u is of the form 4m +1 and each prime factor of v is of the form 4m+3, he stated that E(n) = 0 unless v is a square, and then E(ri) =T(U).
Jacobi12 proved the identity
(6) (l+x+x3+ . . . +xk(k+w+ . . .)4 = l+<r(3)a;+ . . . +ff(2n+I)xn+ ....
A. M. Legendre13 proved (1).
G. L. Dirichlet14 noted that the mean (mittlerer Werth) of cr(ri) is 7r2n/6 -1/2, that of r(n) is log n+2C, where C is Euler's constant 0.57721. . . . He stated the approximations to T(ri) and ^(n), proved later17, without obtaining the order of magnitude of the error.
Dirichlet15 expressed m in all ways as a product of a square by a complementary factor e, denoted by v the number of distinct primes dividing c, and proved that S2"=r(w).
Stern160 proved (2) by expanding the logarithm of (1). If C'n is the number of all combinations with repetitions with the sum n,
Let S(ri) be the sum of the even divisors of n. Then, by (1),
., S(0)=2n.
10Fundamenta Nova, 1829, § 66, (7); Werke, 1, 237. Jour, fiir Math., 21, 1840, 13; French transl., Jour, de Math, 7, 1842, 85; Werke, 6, 281. Cf. Bachmann,0 pp. 31-7.
ll/Wd., §40; Werke, 1, 1881, 163.
"Attributed to Jacobi by Bouniakowsky19 without reference. See Legendre (1828) and Plana (1863) in the chapter on polygonal numbers, vol. 2.
"Theorie des nombres, ed. 3, 1830, vol. 2, 128.
14Jour. fur Math., 18, 1838, 273; Bericht Berlin Ak., 1838, 13-15; Werke, 1, 373, 351-6.
™Ibid., 21, 1840, 4. Zahlentheorie, § 124.
^Ibid., 177-192.