328 History of the Theory of Numbers. [Chap, xi 
To obtain the number Hoin) of secondary numbers ^n, replace square roots 
by cube roots in the /,. We have 
ffi(n)+Hi([|:,])+/fi([|]) + . . . =n, H2{n)+Ho{]^^ + . . . =n, 
and similarly for Hi,_i{n) given by (2) below. 
J. Grolous" considered the probability R^ that a number be divisible 
by at least one of the integers Qi,. . ., Qk, relatively prime by twos, and 
showed that 
Chr. Zeller''" modified Dirichlet's^ expression for h. The sums 
,=iLs J ,=iLs + aJ 
are equal. The sum of the terms of the second with s>fjL = [\/p] equals 
the excess of the sum of the first n terms of the first over fx^ or ju^ — 1 , the 
latter in the case of numbers between fi^ and m^+m- Hence we may abbre- 
\iate the computation of h. 
E. Cesaro^ obtained Dirichlet's^'^ results and similar ones. The mean 
(p. 174) of the number of decompositions of A^ into two factors having p as 
their g. c. d. is 6(log N)/(p~Tr^). The mean (p. 230) of the number of 
di\isors common to two positive integers n, n' is 7rV6, that of the sum of 
their common di\isors is 
ilog, nn'+2C-Y^+i, 
where C = 0.57721 .... The sum of the inverses of the nth powers of two posi- 
tive integers is in mean ^^+2) where s" is defined by (12) of Ch. X. 
E. Cesaro^ 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 n", if (a, b) and (6, a) are both counted when a^^b; 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. 
Cesaro^" noted that the probability that an integer has no divisor > 1 
which is an exact rth power is l/f(r). 
L. Gegenbauer^° proved that the number of integers ^ x and divisible 
by no square is asymptotic to Gx/tt", with an error of order inferior to 
\/x- He proved the final formulas of Bougaief.^ 
'Bull. Sc. Soc. Philomatique de Paris, 1872, 11(>-128. 
'"Nachrichten Gesell. Wiss. Gottingen, 1879, 265-8. 
8M6m. Soc. R. Sc. de Li^ge, (2), 10, 1883, No. 6, 175-191, 219-220 (corrections, p. 343). 
•Annali di mat., (2), 13, 1885, 235-351, "Excursions arith. 4 I'lnfini." 
•"Nouv. Ann. Math., (3), 4, 1885, 421. 
I'Denkschr. 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, M6m. 2, 
p. 110; E. Landau, Bull. Soc. Math. France, 33, 1905, 241. See Gegenbauer,".'" Ch. X. 
