Chap. V] EulER's (^-FUNCTION. 115 
where the summations extend over the combinations of 6,. . ., co taken 
1, 2, . . ., at a time, while Aq is a positive integer <A for which ^Aq+C is 
divisible by A, and [x] is the greatest integer ^x. We thus derive the 
approximation stated by Legendre.^ Taking A = l, C = (p. 420), we see 
that the number of integers ^n, which are divisible by no one of the dis- 
tinct primes 0, X, . . . , co is 
A. von Ettingshausen^ reproduced without reference Euler's^ proof of 
(3) and gave an obscurely expressed proof of (4) . Let A'' = p'g" . . . , where 
p, q,. . . are distinct primes. Consider first only the divisors d = p^q', where 
/i>0, v>0, so that d involves the primes p and q, but no others. By (3), 
^(d)=d(i-^) (i-l), J^ |^py=(p+p'+. . .+r)(g+. . .+2"), 
S(^(pY) = (p"-i)(/-i). 
Similarly, S0(p'') =p"— 1. In this way we treat together the divisors of N 
which involve the same prime factors. Hence when d ranges over all the 
divisors of N, 
S<^(d) = l+S(p«-l) + S(p»-l)(g^-l)+ S (p»-l)(5''_l)(r-_l) + ... 
P P.Q P. 3. r 
=n]i+(p"-i)J=np"=iv, 
p 
where the summation indices range over the combinations of all the prime 
factors of N taken 1, 2, . . .at a time. [Cf, Sylvester .^^] 
A. L. Crelle^ considered the number Zj of integers, chosen from rii, . . . ,na, 
which are divisible by exactly j of the distinct primes Pi, . . ., Pm', and the 
number Sy of the integers, chosen from rii, . . . , n^, which are divisible by at 
least j of the primes Pi. Then 
Z1 + Z2+. . .+Zm = Si-S2 + Ss- . . .=tS^. 
Let V be the number of the integers rii, . . . , n^ which are divisible by no one 
of the primes Pi. Then 
a^'Ezi+v, p = a-Si+S2— . . .=PSrr,. 
In particular, take nj, ..., ria to be 1, 2,. .., iV, where N = p''q^r^ . . ., and 
take Pi,...,Pm to be p, g, r, . . . . Then 
N N ^ N N , N , 
P q pq pr ' pqr 
cf>{N)=N-s,-{-S2- . . . =n(i--^ 0"-) • • •• 
He proved (1) for 5 = 0", where a is a prime not dividing A (p. 40). By 
Euler's^ table there are B({)(A) integers <AB and prime to A. In Euler's 
^Zeitschrif t fur Physik u. Math, (eds., Baumgartner and Ettmg8hauaen),Wien, 5, 1829, 287-292. 
*Abh. Akad. Wiss. Berlin (Math.), 1832, 37-50. 
