Chap. X] SuM AND NuMBER OF DiVISORS. 281 
by the relatively prime numbers a,h, c,. . . is 
Waring noted that o-(a|8) = ao-(/3) + (sum of those divisors of jS which are not 
divisible by a) . Similarly, 
<T(a^y . . . ) = aai^y . . . ) + (sum of divisors of 187 . . . not divisible by a) 
= a(3a{yd . . .) + (sum of divisors of JS7. . . not divisible by a) 
+a(sum of divisors of 76 . . . not divisible by jS), 
etc. Again, (r^'^(a/3)=ao-^"(i8) + (sum of divisors of ^ divisible by I but not 
by a). The generalization is similar to that just given for a. 
C. G. J. Jacobi^^ proved for the series s in (1) that 
00 
s^ = l-3x-\-5x^-7x^+...= S (-l)"(2n+l) a;"'"+^)/2. 
n=0 
Jacobi^^ considered the excess E{n) of the number of divisors of the 
form 4w + l of n over the number of divisors of the form 4m +3 of n. If 
n = 2^uv, 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 y is a square, 
and then E{n) =t{u). 
Jacobi^^ proved the identity 
(6) {l+x-{-x^-{- . . . +a;'(^+i)/24- . . .y = l-\-a(3)x+ . . . +(r(2n+lK+ .... 
A. M. Legendre^^ proved (1). 
G. L. Dirichlet^^ noted that the mean (mittlerer Werth) of (T{n) is x^n/6 
— 1/2, that of T{n) is log n+2C, where C isEuler's constant 0.57721. . . . 
He stated the approximations to T{n) and \pin), proved later^'^, without ob- 
taining the order of magnitude of the error. 
Dirichlet^^ expressed m in all ways as a product of a square by a com- 
plementary factor e, denoted by v the number of distinct primes dividing e, 
and proved that 22" = T(m). 
Stern^^" proved (2) by expanding the logarithm of (1). If C"„ is the 
number of all combinations with repetitions with the sum n, 
(T(n)=nCn-C\ain-l)-C'2(T{n-2)- . . .. 
Let S{n) be the sum of the even divisors of n. Then, by (1), 
S{2n)=Si2n-2)-\-S{2n-4:)-S{2n-10)-Si2n-U)-\- . . ., S{0)=2n. 
"Fundamenta 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,« pp. 31-7. 
"Zfeid., §40; Werke, 1, 1881, 163. 
i^Attributed to Jacobi by Bouniakowsky" without reference. See Legendre (1828) and 
Plana (1863) in the chapter on polygonal numbers, vol. 2. 
"Th^orie des nombres, ed. 3, 1830, vol. 2, 128. 
"Jour, fiir Math., 18, 1838, 273; Bericht Berlin Ak., 1838, 13-15; Werke, 1, 373, 351-6. 
^'Ibid., 21, 1840, 4. Zahlentheorie, § 124. 
i5»76id., 177-192. 
