CHAP. V] ETJLER'S ^-FUNCTION. 127 E. Ces£ro67 proved that, if / is any function, ^ af( = - xnp(n^ F(n) where d ranges over the divisors of n. For/=<£, we have F(x) = x and obtain Liouville's29 first formula. By the same specialization (p. 64) of another formula (given in Chapter X on sums of divisors61), Ces&ro derived the final formula of Liouville.2S If (n, j) is the g. c. d. of n and j, then (p. 77, p. 80) If (p. 94) p is one of the integers a, 0, . . . :gn and prime to n, S0(a)F(a)=S(?(a)/(a), F(x)=2f(d), G(p)m2g(p a where d ranges over the divisors of x. For g(x) = 1, this gives where (p. 96) $(n, x) is the number of integers ^x and prime to n. Ces&ro (pp. 144-151, 302-3) discussed and modified Perott's53 proof of his first formula, criticizing his replacement of [n/k] by n/k for n large. He gave (pp. 153-6) a simple proof that the mean67 of <t>(ri) is 6n/7r2 and reproduced the proofs by Dirichlet21 and Mertens,36 the last essentially the same as Perott's. For f(m) = equal asymptotically (pp. 167-9) f W/f (m+1), (6 log n)/7r2, f (m+1), log n. As a corollary (p. 251) to Mansion's41 generalization of Smith's theorem we have the result that the determinant of order n2, each element being 1 or 0 according as the g. c. d. of its two indices is or is not a perfect square, equals ( — l)a+b+ "-t where paqb . • • is the value of nl expressed in terms of its prime factors. Ces^ro58 considered any function F(xt y} of the g. c. d. of x, y, and the determinant AM of order n having the element F(uit Uj) in the ith row and jth column, where uit . . . , un are integers in ascending order such that each divisor of every u{ is a u. Employing the function /z(n) [see Ch. XIX], he noted that . Soc. R. Sc. de Ltege, (2), 10, 1883, No. 6, 74. "Atti Reale Accad. Lincei, (4), 1, 1884-5, 709-711.