116 History of the Theory of Numbers. (Chap, v 
notation, a{kA + l), a{kA-\-a),. . ., a{kA-{-03) give all the numbers between 
kaA and {k-\-\)aA which are divisible by a and are prime to A. Taking 
A: = 0, 1,. . ., a°~^ — 1, we see that there are exactly a"~V(^) multiples of a 
which are <AB and prime to A. Hence 
0(a"^) =aXA) -o^-VU) =(t>{a'')(f>{A). 
F. Minding^ proved Legendre's formula (5). The number of integers 
^n, not divisible by the prime 0, is n — [n/d]. To make the general step 
by induction, let Pi, . . . , Pk be distinct primes, and denote by (5; pi, . . . , p^) 
the number of integers ^ 5 which are divisible by no one of the primes pi, . . . , 
Pk' Then, if p is a new prime, 
(B; pi, . . . , Pk, p) = {B;pu..., Pk) - ([B/p] ; Pi, . • • , Pk)- 
The truth of (4) for the special case N = p — 1, where p is a prime, follows 
(p. 41) from the fact that ((){d) numbers belong to the exponent d modulo p 
if d is any divisor of p — 1. 
N. Druckenmiiller^*^ evaluated (f>{b), first for the case in which 6 is a 
product cd. . .kl oi distinct primes. Set h=^l and denote by \f/{h) the num- 
ber of integers <b having a factor in common with 6. There are l\p{^) 
numbers < b which are divisible by one of the primes c, . . . , k, since there 
are \p{P) in each of the sets 
l,2,...,/3; ^+1,...,2^; ...; (i-l)/3+l,. . ., Z/3. 
Again, I, 21,..., pi are the integers <b with the factor I. Of these, 0(j3) 
are prime to jS, while the others have one of the factors c, . . . , k and occur 
among the above lxl/{^). Hence xl/{b)=l\l/i^)-\-<i>iP). But i/'O3)+0(/3)=/3. 
Hence 
</,(6) = a-l)(A(^) = (c-l)...(Z-l). 
Next, let 6 be a product of powers oi c, d,. . ., I, and set b = L^, ^ = cd. . .1. 
By considering L sets as before, we get 
E. Catalan^^ proved (4) by noting that 
2(/,(py. . .)=n]i+(/)(p)+ . . . +<f>(p'')\ =np»=Ar, 
where there are as many factors in each product as there are distinct prime 
factors of N. 
A. Cauchy^^ gave without reference Gauss'^ proof of (1). 
E. Catalan^^ evaluated <t){N) by Euler's^ second method. 
C. F. Arndt^"* gave an obscure proof of (4), apparently intended for 
Catalan's. ^^ It was reproduced by Desmarest, Th^orie des nombres, 1852, 
p. 230. 
•Anfangsgriinde der Hoheren Arith., 1832, 13-15. 
»oTheorie der Kettenreihen . . .Trier, 1837, 21. 
"Jour, de Mathdmatiques, 4, 1839, 7-8. 
i»Compte8 Rendus Paris, 12, 1841, 819-821; Exercices d'analyse et de phys. math., Paris, 2, 
1841, 9; Oeuvres, (2), 12. 
"Nouv. Ann. Math., 1, 1842, 466-7. 
"Archiv Math. Phys., 2, 1842, 6-7. 
