Chap. V] EulEE's (^-FUNCTION. 127 
E. Ces^ro^^ proved that, if / is any function, 
X^i^Xx^Fin), F(n)^S/(d), 
n = ll X 71=1 
where d ranges over the divisors of n. For / = 0, we have F(x)=x and obtain 
Liouville's^^ first formula. By the same specialization (p. 64) of another 
formula (given in Chapter X on sums of divisors^^), Cesaro derived the 
final formula of Liouville.^^ If (n, j) is the g. c. d. of n and j, then (p. 77, 
p. 80) 
Mn, j) =S#Q), X-^ = h:d<t>{d), X4>{n, j) =S(^(d)0(-). 
If (p. 94) p is one of the integers a, /3, . . . ^ n and prime to n, 
S^(a)F(a) =SG(a)/(a), /^(a:)^S/(d), G(p)^S^(pa), 
a 
where d ranges over the divisors of x. For g{x) = l, this gives 
S/(a)0(n, n/a)=SF(a), 
a 
where (p. 96) </)(n, x) is the number of integers ^x and prime to n. Cesaro 
(pp. 144-151, 302-3) discussed and modified Perott's^^ 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 mean^^ of <^(n) is Qn/w^ and reproduced 
the proofs by Dirichlet^^ and Mertens,^^ the last essentially the same as 
Perott's. For Kw) = l + l/2"^+l/3"+. . ., 
s4r(^>l), 2i 2-i-(m>l), 2-^ 
a"* ' '' a' a'"</)(a) ' '' 0(a) 
equal asymptotically (pp. 167-9) 
f(m)/r(m+l), (6 1og7i)/7r^ r(m+l), log n. 
As a corollary (p. 251) to Mansion's^^ generalization of Smith's theorem we 
have the result that the determinant of order n^, each element being 1 or 
according as the g. c. d. of its two indices is or is not a perfect square, equals 
( — 1)"+^+- , where pV- • • is the value of n\ expressed in terms of its prime 
factors. 
Ces^ro^* considered any function F{x, y) of the g. c. d. of x, y, and the 
determinant A„ of order n having the element F{Ui, u/) in the ith. row and 
ith column, where Ui,...,Un are integers in ascending order such that each 
divisor of every Ui is a u. Employing the function ix{n) [see Ch. XIX], he 
noted that 
i nO^)F(u„ud=f{u,) or 0, 
"M6m. Soc. R. Sc. de LiSge, (2), 10, 1883, No. 6, 74. 
"Atti Reale Accad. Lincei, (4), 1, 1884-5, 709-711. 
