136 History of the Theory of Numbers. [Chap, v 
Then 
Generalizing /x(s), let )u-*^(s) be zero if the expansion of the product 
n(l— p)*", extended over all primes p, does not contain a term equal to s, 
but let it equal the coefficient of s if s occurs in the expansion. Then 
i?,(n)=SdM'"Q) 
The 7i-rowed determinant in which the element in the rth row and sth 
column is F„_i(5), where 5 is the g. c. d. of r, s, is proved equal to F^(l) 
F^(2) . . .Fm{n), a generaUzation of Smith's^^ theorem. Finally, 
isf.,,Q)f_.(d)=siF,(d), 
the right member being T{n), 20(c?)/c?, lla{d)/d for r = 0, 1, —1. 
G. Landsberg^^" gave a simple proof of Moreau's"^ formula for the 
number of circular permutations. 
L. Carlini^^ proved Dirichlet's^^ formula by noting that 
(8) n(x''-l)=0 
has unity as an n-f old root, while a root 7^ 1 of x'' — 1 is a root of [n/h] factors 
x"* — 1. Hence the 4){h) primitive roots of x^ = \ furnish <l>{h)[n/h] roots 
of (8). 
M. Lerch^^ found the number N of positive integers '^m which have no 
one of the divisors a, 6, . . . , k, I, the latter being relatively prime in pairs 
and ha\'ing m as their product. Let F{x) = 1 or 0, according as x is frac- 
tional or integral. Let L = ab. . .k. Then [Dirichlet^^] 
^^m(Z-l)^ 
,!/©-©-(-■) (-i) 
L 
E. Landau^^ proved that the inferior limit for a:= 00 of 
-<f>{x) log. log, X 
X 
is e~^, where C is Euler's constant. Hence <^(j) is comprised between this 
inferior limit and the maximum x — 1. 
R. Occhipinti^°° proved that, if aj is an nth root of unity, and if c?,i, • ■ • , 
dat are the divisors of i, 
n|s<^(di,)+a,S<^(d2i)+. . .+a/-4V(0| = i(-l)"-'n(n+l)n"-2. 
j-lU-l i-l i-l J 
••"Archiv Math. Phys., (3), 3, 1902, 152-4. "Periodico di Mat., 17, 1902, 329. 
"Prag Sitzungsber., 1903, II. "Archiv Math. Phys., (3), 5, 1903, 86-91. 
"«Periodico di Mat., 19, 1904, 93. Handbuch,"' I, 217-9. 
