Chap. V] EulEE's (^-FUNCTION. 129 
Ces^ro" employed F(n)=S/(d), G(n)='Zg{d), where d ranges over the 
divisors of n, and proved that 
G(l) G{2) ... G{n) 
G{\) F(l,l) F(l,2) ... F(l,n) -_j(i)..j(^)p'(^), 
G{n) F{n,l) F{n,2) Fin, n) 
In particular, if F{n) is the number of divisors of n and if G{n) is the number 
of prime divisors of n, the determinant, apart from signs, equals the number 
of primes ^ n. 
E. Cesaro^^ wrote (a, b) for the g. c. d. of a, h. If F{n) =S/(d), where d 
ranges over the divisors of n, then 
XF\(n,i)\=i:f(d)N/d. 
In particular, if /,(n) is the number of irreducible fractions ^e of denomi- 
nator n. 
IXn)=i:[j]n{d), S7,(d) = [ne]. 
The last formula, due to Laguerre,^^ follows by inversion (Ch. XIX), and 
directly from the fact that I^d) is the number of the first [ne] integers which 
with n have the g. c. d. n/d. The number of irreducible fractions ^ e of 
denominator ^n is $e(^) =-f «(!)+••• +-^.(^). We have 
00 \n/j] O, 
^M = SmO') S [ie], Imi $,(n)/n2 = -2 (€>0), 
j = l 1=1 n=oo T 
due to Sylvester^^ for € = 1. Let (^^g^(n) be the sum of the j'th powers of 
the numerators of the irreducible fractions < e of denominator n. Set 
Then 
$« in) = S </)(? ii) , sXn) = S ^^ 
1=1 »-i 
i=l LzJ i.i 
which generalizes the two formulas of Sylvester .^^ Also, 
$^;^ (w) = — — — — — , asymptotically. 
TT V-\-l V-\-2 
Ces^ro^^" factored determinants of the tj^e in his paper,^^ the function F 
now being such that Fixy)/ \Fix)Fiy)\ is a function of the g. c. d. oi x, y. 
L. Gegenbauer®^'' gave a complicated theorem involving several general 
functions, special cases of which give Sylvester's^^ two summation formulas. 
"Nouv. Ann. Math., (3), 5, 1886, 44-47. 
"Annali di Mat., (2), 14, 1886-7, 143-6. 
""Giornale di Mat., 25, 1887, 18-19. 
«5fcSitzungsber. Ak. Wiss. Wien (Math.), 94, 1886, II, 757-762. 
