HlSTOBY OF THE THBOEY OF NtTMBEKS. [CHAP. XIX
-v,
B Merry7 gave a proof by noting that, if d is any divisor of m, and if q of the prime factors of m occur to the same power in d as in m, then f(d) occurs once in F(m), fl times in 2F(ro/o), g(g-l)/2 times in 2F(m/a6), etc. Thus the coefficient of /(d) in .(2) is
if g>0, but is unity if 0=0, i. e., if d=m. This proof is only another way of stating Dedekind's proof .
R. Dedekind8 gave another form and proof of his theorems. Let
where vi ranges over the positive terms of the expanded product and -v2 over the negative terms. A simple proof shows that, if v is any divisor <m of m, there are as many terms *>i divisible by *> as terms v2 divisible by v.
ThuS
imply, respectively,
/(m) -2FW -
Liouville8* wrote F(n) =S/(n/DM), where D ranges over those divisors of aaV. . . for which D" divides n. Then /(n) =F(n) -2F(n/af) +2 E. Laguerre9 expressed (2) in the form
(3) /fn»
where d ranges over the divisors of m. Let
2/(n)T=2F(nx»,
n-l i~~ X n-1
whence F(m)=2/(d). For m=Hpa, where the pjs are distinct primes, let /(m)=n/(pa)J and/(pm)-pw-1(p-l). Then
Fm =nl+/(p)+ • • • +/(Pa""1)) =npa = m.
The hypotheses are satisfied if / is Euler's function </>. This discussion deduces S0(d)=m from the usual expression of type (3) for <£(m), rather than the reverse as claimed.
N. V. Bougaief10 proved (1).
F. Mertens11 defined /i(n) and noted that S/*(d)=0 if n>l, where d ranges over the divisors of n. _ __
7Nouv Ann Math 16, 1857, 434.
•Dirichlet'a Zahlentheorie, mit Zusfitzen von Dedekind, 1863, §138; cd. 2, 1871, p. 356; ed, 4,
1894, p. 360.
•ojour. de Math., (2), 8, 1863, 349. •Bull. Soc. Math. France, 1, 1872-3, 77-81.
"Mat. Sbornik (Math. Soc. Moscow), 6, 1872-3, 179. Cf. Sterneck.19 "Jour, fttr Math., 77, 1874, 289; 78, 1874, 53.