Chap. V] EuLER's 0-FuNCTION. 119 
G. L. Dirichlet^^ added equations (4) for iV = n, . . ., 2, 1, noting that, 
if s^n, 4>(s) occurs in the new left member as often as there are mul- 
tiples ^n oi s. Hence 
i\-']<f>{s)=Un'+n). 
s=lLsJ 
The left member is proved equal to XxJ/in/s], where 
It is then shown that \p{n) —Zn^/ir^ is of an order 6i magnitude not exceed- 
ing that of n\ where 2>5>7>1,7 being such that 
8=2 S^ 
P. L. Tchebychef^^ evaluated 0(n) by showing that, if p is a prime not 
dividing A, the ratio of the number of integers ^ pAN which are prime to A 
to the number which are prime to both A and p is p:p — l. 
A. Guilmin^^ gave Crelle's^^ argument leading to 0(Z). 
F. Landry ^^ proved (3). First, reject from 1, . . ., iV the N/p multiples 
of p; there remain A^(l — 1/p) numbers prime to p. Next, to find how many 
of the multiples q, 2q, . . . , N of q are prime to p, note that the coefficients 
1, 2, . . ., N/q contain N/q-{l — l/p) integers prime to p by the first result, 
applied to the multiple N/q of p in place of N. 
Daniel Augusto da Silva^^ considered any set S of numbers and denoted 
by S{a) the subset possessing the property a, by S{ab) the subset with the 
properties a and b simultaneously, by {a)S the subset of numbers in S 
not having property a; etc. Then 
{a)S = S-S{a)=S\l-{a)\, 
symbolically. Hence 
(ha)S = {b)\(a)S\=S\l-{a)\\l-{h)\, 
{. . .cba)S^S\l-{a)\\l-{b)\ \l-{c)\ . . .. 
A proof of the latter symbohc formula was given by F. Horta.^^" 
With Silva, let *S be the set 1, 2, . . . , n, and let A, ^, . . . be the distinct 
prime factors of n. Let properties a, 6, ... be divisibility by A,B,. . .. Then 
there are n/A terms in >S(a), n/{AB) terms in S{ab), . . ., and <f){n) terms in 
( . . .cba)S. Hence our symbolic formula gives 
*(«)=»(l-i)(l-|). 
"Abhand. Ak. Wiss. Berlin (Math.), 1849, 78-81; Werke, 2, 60-64. 
^^Theorie der Congruenzen, 1889, §7; in Russian, 1849. 
«Nouv. Ann. Math., 10, 1851, 23. 
^^Troisieme mlmoire siir la th^orie des nombres, 1854, 23-24. 
'^Proprietades geraes et resoluQao directa das Congruencias binomias, Lisbon, 1854. Report 
on same by C. Alasia, Rivista di Fisica, Mat. e Sc. Nat., Pavia, 4, 1903, i3-17; reprinted 
in Annaes Scientificos Acad. Polyt. do Porto, Coimbra, 4, 1909, 166-192. 
""Annaes de Sciencias e Lettras, Lisbon, 1, 1857, 705. 
