Chap. V] EuLER's (^-FUNCTION. 121 
where m in S' ranges only over the positive odd integers. The final fraction 
equals x-\-Sx^-{-5x^+ .... From the coefficient of x^ in the expansion of 
the third sum, we conclude that, if n is even, 
where d ranges over all the divisors of n. Let 5i range over the odd values 
of 5, and 82 over the even values of 5; then 
0-0- 
the value n/2 following from (4). Another, purely arithmetical, proof is 
given. Finally, by use of (4), it is proved that, if s>2, 
n=l /fr n=l/t 
A. Cayley3° discussed the solution for N of <^(iV) = iV^ Set N = a^b^ ..., 
where a, 6, . . . are distinct primes. Multiply 
l + (a-l) \a\+a{a-l) \a^\ + ... +a''-\a-l) {a"} + . . . 
by the analogous series in 5, etc. ; the bracketed terms are to be multiplied 
together by enclosing their product in a bracket. The general term of the 
product is evidently 
Hence in the product first mentioned each of the bracketed numbers which 
are multiplied by the coefficient N' will be a solution N of <f){N)=N'. We 
need use only the primes a for which a — 1 divides N', and continue each 
series only so far as it gives a divisor of N' for the coefficient of a"~^(a — 1). 
V. A. Lebesgue^^ proved 4){Z)=n4>{z) as had Crelle^^ and then 4>{z) 
=n(pi — 1) by the usual method of excluding multiples of pi, . . . , p„ in turn. 
By the last method he proved (pp. 125-8) Legendre's (5), and the more 
general formula preceding (5). 
J. J. Sylvester^^ proved (4) by the method of Ettingshausen,' using (2) 
instead of (3) . By means of (4) he gave a simple proof of the first formula 
of Dirichlet;^^ call the left member u^', since [n/r] — [(n — l)/r] = l or 0, 
according as n is or is not divisible by r, 
v^^J^ w(n+l) 
The constant c is zero since Ui = l. He stated the generalization 
2{*(i')(l-+2-+... + [?]'")}^ 
r+2'-+...+n'". 
He remarked that the theorem in its simplest form is 
"London Ed. and Dublin Phil. Mag., (4), 14, 1857, 539-540. 
"Exercicea d'analyse niim^rique, 1859, 43-45. 
"Quar. Jour. Math., 3, 1860, 186-190; CoU. Math. Papers, 2, 225-8. 
