328 HISTORY or THE THEORY OF NUMBERS. [CHAP, xi To obtain the number H2(ri) of secondary numbers ^n, replace square roots by cube roots in the <£. We have and similarly for H^i(n) given by (2) below. J. Grolous7 considered the probability Rk that a number be divisible by at least one of the integers Qly..., Qk, relatively prime by twos, and showed that Rn = 7j-+7C(l-R1)+- • .+^d-«n-1). wi ^2 yn Chr. Zeller7a modified Dirichlet's4 expression for h. The sums p rn "| p-1 r n 1 S --ah 2-7— ,=iLs J ^iLs+aJ are equal. The sum of the terms of the second with s>ju==[v/p] equals the excess of the sum of the first M terms of the first over ^ or ju2 — 1, the latter in the case of numbers between ju2 and M2+ju. Hence we may abbreviate the computation of h. E. Ces&ro8 obtained Dirichlet's3'4 results and similar ones. The mean (p. 174) of the number of decompositions of N into two factors having p as their g. c. d. is 6(log N)/(p2ir2). The mean (p. 230) of the number of divisors common to two positive integers n, n' is 7^/6, that of the sum of their common divisors is flog, nn'+2C-§+J, where C = 0.57721.... The sum of the inverses of the nth powers of two positive integers is in mean f (n+2) where f is defined by (12) of Ch. X. E. Cesaro9 proved the preceding results on mean values; showed that the number of couples of integers whose 1. c. m. is n is the number of divisors of n2, if (a, 6) and (6, a) are both counted when a^fr; found the mean of the 1. c. m. of two numbers; found the probability that in a random division the quotient is odd, and the mean of the first or last digit of the quotient; the probability that the g.c.d. of several numbers shall have specified properties. Ces£ro9a noted that the probability that an integer has no divisor > 1 which is an exact rth power is l/f(r). L. Gegenbauer10 proved that the number of integers :g x and divisible by no square is asymptotic to 6z/7r2, with an error of order inferior to -\/x- He proved the final formulas of Bougaief.6 7Bull. Sc. Soc. Philomatique de Paris, 1872, 119-128. 70Nachrichten Gesell. Wiss. Gottingen, 1879, 265-8. 8M6m. Soc. R. Sc. de Lifege, (2), 10, 1883, No. 6, 175-191, 219-220 (corrections, p. 343). »Annali di mat., (2), 13, 1885, 235-351, "Excursions arith. al'infim." ""Nouv. Ann. Math., (3), 4, 1885, 421. 10Denkschr. Akad. Wien (Math.), 49,1, 1885, 47-8. Sitzungsber. Akad. Wien, 112, II a, 1903, 562; 115, II a, 1906, 589. Cf. A. Berger, Nova Acta Soc. Upsal., (3), 14, 1891, Mem. 2, p. 110; E. Landau, Bull. Soc. Math. France, 33, 1905, 241. See Gegenbauer,72,79Ch. X.