446 History of the Theory of Numbers. [Chap, xix 
of rii, . . . , 7J,. Then, if a ranges over those divisors d which are divisible by 
no one of ^i, . . . , v^', chosen from nj, . . . , n^, 
The left member equals F{m, N, e) constructed for the numbers other 
than Vi,..., V,' of the set ni, . . . , /z^. For s' = s, we have 
I 
f(m, N, e) =F(m, N, e) -Si^([f ]' ^' ^n^ 
+ . 
The latter becomes the series in Bachmann^° when m=cD, N = 0, € = 1, 
while rii, n2, . . . are primes. 
F. Mertens^^ considered (7(n) =ju(l)H-/i(2) + . . . +M(n) and proved that 
l.o)[f]=.g)+.(l)+.. .+.©-..(.). 
<rin) = 2a{g) - X n{rMs) f^l , g = [V^]. 
r. s=i LrsJ 
By means of a table (pp. 781-830) of the values of a{n) and /x(n) forn< 10000, 
it is verified that \(7{n)\< \/n for 1< n< 10000. 
D. von Sterneck^^ verified the last result up to 500 000, and for 16 larger 
values under 5 million. 
A. Berger" noted that, if g{m)g{n)=g{mn), ^(1) = 1, 
i:n{d)gid)=U{l-gip)] (n>l), 
where d ranges over all divisors of n, p over the prime divisors of n. If 
Xg{m) is absolutely convergent, 
sM%W=n{i-^(p)), 
where p ranges over all primes. 
D. von Sterneck^^ noted that, if 6ix) ^ 1 for every x and if 
|j.w|<^6-H|/(.)-/[g-/[=]-/[g|. 
In particular, |2iu(/c)|<8+?z/9. 
D. F. Seliwanov'-^ gave Dedekind's formula with appHcation to <t>{n). 
H. von Koch^^" defined n{k) by use of infinite determinants. 
"Sitzungsber. Ak. Wiss. Wien (Math.), 106, II a, 1897, 761-830. 
»Ibid., 835-1024; 110, Ila, 1901, 1053-1102; 121, Ila, 1912, 1083-96; Proc. Fifth Intern. Con- 
gress Math., 1912, I, 341-3. 
"Ofversigt Vctenskap.s-Akad. Forhand., Stockholm, 55, 1898, 579-618. 
"Monatshcfte Math. Phys., 9, 1898, 43-5. 
"Math. Soc. St. Paersbourg, 1899, 120. 
""Ofversigt K. Vetensk.-Akad. Forhand., Stockholm, 57, 1900, 659-68. 
