448 History of the Theory of Numbers. [Chap, xix 
R{i), — 1 if a product of an odd number, if x is di\'isible by the square of a 
prime of R{i). Let [771] denote a complete set of residues ?^0 of complex 
integers modulo m. Then the sum of the values of fix) for all complex 
integers x relatively prime to a given one n, which are in [m], equals 
2jLi(c?)2/(d'x), where d ranges over all divisors of n in [m], and x ranges 
over {rn/d}. This is due, for the case of real numbers, to Nazimov^" of 
Chapter V. Again, 2/i((i) = l or according as norm n is 1 or >1. Also 
2/(d) =F(7?) miplies Xfi{d)F(n/d) =f{n). 
J. C. Kluyver^ employed Kronecker's^" identity for special functions / 
and obtained known results like 
2 cos — =/x(n), n 2sin — = e^^''\ 
n n 
where v ranges over the integers <Ji and prime to n, while y{n) is Bouga- 
ief's^^ function p{n). 
P. Fatou^ noted that Merten's a{n) does not oscillate between finite 
limits. E. Landau^^ proved that it is at most of the order of ne', where 
t = — aVlog n. Landau^® noted that Furlan" made a false use of analysis 
and ideal theory to obtain a result of Landau's on Merten's^^ (r(n). 
0. ■Meissner^^" emploj-ed primes p,, q,. For n=Up'i set Z(n)=nei''> 
and Z2{n) = Z\Z{n)\. Then Z{?i) = n only if n is Up.^i or 16 or Hjp'iqii. 
Next, Z2(n)=n in these three cases and when the exponents e, in n are 
distinct primes; otherwise, Z2(n)<n. We have [1/Z{n)]=fx'^{n). 
R. HackeP^ extended the method of von Sterneck^^ and obtained vari- 
ous closer approximations, one^^ being 
xe{k) 
<^+io2+|2/g]-2/[g|, 
26 
where a = l, 6, 10, 14, 105; 6 = 2, 3, 5, 7, 11, 13, 385, 1001. 
W. Kusnetzov-^^ gave an analytic expression for ;u(n). 
K. Knopp^^° of Ch. X gave many formulas involving n{n). 
A. Fleck''"" generahzed n{m)=iJLi{m) by setting 
M*(m)=n(-l)»'(f), m=Upr. 
,=1 \a,/ ,=1 
Using the zeta function (12) of Ch. X, and ^^ of Fleck^-^ of Ch. V, we have 
y r^\ r \ V ^k-\{m) . . * ^t(m) ^ (m\ 
i:Hk{d)=fXk-i{m), S — — — =f(s) 2 — — -, <t)k{m) = Xdfik+A-T)- 
dim fn = l ^ m = l ^ dm \"/ 
"Verslag. Wiss. Ak. Wetenschappen, Amsterdam, 15, 1906, 423-9. Proc. Sect. Sc. Ak. Wet., 
9, 1906, 408-14. "Acta Math., 30, 1906, 392. 
«Rend. Circ. Mat. Palermo, 26, 1908, 250. "Rend. Circ. Mat. Palermo, 23, 1907, 367-373- 
"Monatshefte Math. Phys., 18, 1907, 235-240. 
»"Math. Xaturw. Blatter, 4, 1907, 85-6. 
"SitzunKsber. Ak. Wiss. Wien (Math.), 118, 1909, II a, 1019-34. 
'•Sylvester, Messenger Math., (2), 21, 1891-2, 113-120. 
«°Mat. Sbomik (Math. Soc. Moscow), 27, 1910, 335-9. 
'■wSitzungsber. Berlin Math. GeseU., 15, 1915, 3-8. 
