442 History of the Theory of Numbers. [Chap. XIX 
B. Merry' 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 c? as in ???, then f{d) 
occurs once in F{m), q times in ZF{m/a), q{q — l)/2 times in 'LFim/ab), etc. 
Thus the coefficient of f{d) in (2) is 
if 5 > 0, but is unity ii q = 0,i. e.,ii d = m. This proof is only another way of 
stating Dedekind's proof. I 
R. Dedekind^ gave another form and proof of his theorems. Let 
(-i)(-i) 
m(l--)(l-T 1 . . =2j/i-2j/2, 
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 di\'isor 
<m of m, there are as many terms vi di\'isible by v as terms V2 divisible by v. 
Thus 
i:f{v)=F{m), Uf{p)=F{m) 
imply, respectively, 
fim) =2F(.,) -2F(^2), Km) -^^y 
Liou\'ille^ wTote F(n) =Tif{n/D''), where D ranges over those di\4sors of 
n = a"}/ ... for which D" divides n. Then 
f{n) =F{n) -XFin/a") +lF{n/a''b'') -.... 
E. Laguerre^ expressed (2) in the form 
(3) /(m)=2M(|)F(d), 
where d ranges over the divisors of m. Let 
2/(n)T^=SF(n)x^ 
n = l L—X n=l 
whence F{m) ='Zf(d). For m^Up", where the p's are distinct primes, let 
f{m) =n/(p»), and/(p") =p''-np-l). Then 
F{m) =njl+/(p) + . . . +f{p''-')] =Up'' = m. 
The hypotheses are satisfied if / is Euler's function 4>. This discussion 
deduces l,(t){d)=m from the usual expression of type (3) for <j){m), rather 
than the reverse as claimed. 
N. V. Bougaief^° proved (1). 
F. Mertens" defined /x(n) and noted that 2/x(d)=0 if n>l, where d 
ranges over the divisors of n. 
'Nouv. Ann. Math. 16, 1857, 434. 
8Dirichlet's Zahlentheorie, mit Zusatzen von Dedekind, 1863, §138; ed. 2, 1871, p. 356; ed. 4, 
1894, p. 360. 
'"Jour, de Math., (2), 8, 1863, 349. 
•BuU. Soc. Math. France, 1, 1872-3, 77-81. 
'"Mat. Sbomik (Math. Soc. Moscow), 6, 1872-3, 179. Cf. Sterneck." 
"Jour, fur Math., 77, 1874, 289; 78, 1874, 53. 
I 
