130 HISTORY OF THE THEORY OF NUMBERS. [CHAP.V P. S. Poretzky66 gave a formula for the function \[/(m) whose values are the 0(ra) integers <m and prune to m. For the case w = 2-3-5. . .p, where p is a prime, «»)-j 3 U-2 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&ro67 gave a very simple proof of the known fact that which he expressed in words by saying that 0(n) is asymptotic to 6n/7r2 (not meaning that the limit of 0(n)/w *s 6/V2). On the distinction between asymptotic mean and median value, see Encyclopedic des sc. math., I, 17 (vol. 3), p. 347. Ces&ro68 noted that if F(i, j) is a function of the g. c. d. of i, j, then Q=2F(i,j) XiXj (i, y=l,. .., ft) becomes q=2f(i)y? by the substitution yk=xk+Xzk+Xak+ • - M provided F(ri) =S/(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.39 A generalization is obtained by use of SF(et-, tj)x&j9 where the numbers ei, e2, . . . include the divisors of each e. E. Catalan69 proved that, if d ranges over the divisors of N = a"bfi . . ., d [ a E. Busche70 derived at once from Dirichlet's21 formula the result 2 0(s) \P +p + . - . \ - aj—l \X/ \ X / where p(a) = a - [a]. The case n = n' = n" = . . . leads to where x takes all values for which p(n/x)>p(vn/x). If we take n=l and add 0(1) = 1, we get (4) for N = v. Next, S0(x)=rr/62, where x takes all values for which j?X 0-1 ..... ,;,--. ...... •>. 68Math. phys. soc. Kasan, 6, 1888, 52-142 (in Russian). "Comptes Rendus Paris, 106, 1888, 1651; 107, 1888, 81, 426; Annaii di Mat., (2), 16, 1888-9, 178 (discussion with Jensen on terminology). 68Atti Reale Accad. Lincei, Rendiconti, 2, 1888, II, 56-61. 69M<§m. Soc. Sc. LiSge, (2), 15, 1888, No. 1, pp. 21-22; Melanges Math., Ill, No. 222, dated 1882. 70Math. Annalen, 31, 1888, 70-74.