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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
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        + . - . \ -
ajl               \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.