130 History of the Theory of Numbers. [Chap, v 
P. S. Poretzky^^ gave a formula for the function \l/{m) whose values are 
the 4>{7?i) integers <?fi and prime to m. For the case w = 2-3-5. . .p, where 
p is a prime, 
lpi-2 Pi J 
where K is an integer. Application is made to the finding of a prime 
exceeding a given number, and to a generalization of the sieve of Eras- 
tosthenes. 
E. Ces^ro^^ gave a very simple proof of the known fact that 
2 2' 
n-00 n^ T 
which he expressed in words by saying that 0(n) is asymptotic to 6n/7r^ 
(not meaning that the limit of 4>{n)/n is G/tt"). On the distinction between 
asymptotic mean and median value, see Encyclop^die des sc. math., I, 
17 (vol. 3), p. 347. 
Ces^ro^^ noted that if F{i, j) is a function of the g. c. d. of i, j, then 
Q=SF(i, j) XiXj {i, j = l,..., n) becomes q='Lf{i)yi^ by the substitution 
yk = Xk-{-X2k-\-Xsk-\- ■ ■ -, provided F{n) =2/(d), d ranging over the divisors of 
n. Since the determinant of the substitution is unity, the discriminants 
of Q and q are equal. Hence we have the theorem of Smith.^^ A gen- 
eralization is obtained by use of 2F(e„ e^XiXj, where the numbers ei, C2, . . . 
include the divisors of each €. 
E. Catalan^^ proved that, if d ranges over the divisors of iV = a"6^ . . . , 
E. Busche^° derived at once from Dirichlet's^^ formula the result 
S0(x))p(^)+p(^) + ...(=Snn', 
j=l \x/ \x / 
where p(a) =a — [a]. The case n = n' =n" = . . . leads to 
i:4>{x) = {v-\)n^, 
where x takes all values for which p{n/x)>p{vn/x). If we take n = l and 
add </)(!) = 1, we get (4) for N = v. Next, S0(a;) =rr'5", where x takes all 
values for which 
yJ±zi^,Q<yyi (y=i,...,.;,'=i,...,.'). 
r-\-r \xy r r 
6«Math. phys. soc. Kasan, 6, 1888, 52-142 (in Russian). 
"Comptcs Rendus Paris, 106, 1888, 1651; 107, 1888, 81, 426; Annali di Mat., (2), 16, 1888-9, 
178 (discussion with Jensen on terminology). 
•8Atti Rcale Accad. Lincei, Rendiconti, 2, 1888, II, 56-61. 
"M6m. Soc. Sc. Li^ge, (2), 15, 1888, No. 1, pp. 21-22; Melanges Math., Ill, No. 222, dated 1882. 
"Math. Annalen, 31, 1888, 70-74. 
