282 History of the Theory of Numbers. [Chap, x 
Let S'(n) be the sum of the odd di\'isors of n, and C„ be the number of all 
combinations without repetitions with the sum n, so that C7 = 5. Then 
S'in)=nCn-S'{n-l)Ci-S\n-2)C2+ . . ., 
Z)(n) = -D(n-l)-D(n-3)-D(n-6)-..., D{n)=S'{n)-S{n). 
A complicated recursion formula for T(n)is derived from 
\og{{l-x){l-x^y{l-3^)r . .} = - I ^-Tin)x\ 
n=in 
Complicated recursion fonnulas are found for the number of integers 
<m not factors of m, and for the sum of these integers. A recursion 
formula for the sum Sr{n) of the di\'isors ^r of n is obtained by expanding 
log {l-x)(l-x2)...(l-a:'-)l = - S -Sr(n)x". 
n=in 
Jacobi^® proved (1). 
Dirichlet^^ obtained approximations to T(n). An integer s^n occurs 
in as many terms of this sum as there are multiples of s among 1, 2, . . . , n. 
The number of these multiples is [n/s], the greatest integer ^n/s. Hence 
''(")=iG] 
This sum is approximately the product of n by 
£i = logn+C+i+.... 
Hence T{n) is of the same order of magnitude as n log n. 
Let ju be the least integer ^ y/n and set v = [n/ii]. Then if g{x) is any 
function and G{x)=g{l)-\-g{2)-\- . . . +^(x), 
2 r^i^(s)= -.GGu)+s pi^(s)+s Gjr^ii- 
»=:LsJ «=iLsJ «=i LLsJJ 
In particular, if ^(x) = 1, 
«=iLsJ «=iLsJ 
Giving to [n/s] the approximation n/s, we see that 
(7) T(n)=n log,n+(2C-l)n+e, 
where € is of the same order of magnitude as Vn. 
Let pin) be the number of distinct prime factors >1 of ti. Then 2"^"^ is 
the number of ways of factoring n into two relatively prime factors, taking 
"Jour, fur Math., 32, 1846, 164; 37, 1848, 67, 73. 
"Abhand. Ak. Wiss. Berlin, 1849, Math., 69-83; Werke, 2, 49-66. French transl., Jour, de 
Math., (2), 1, 1856, 353-370. 
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