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CHAP. V]                              EULER'S ^-FUNCTION.                                    121
where m in S' ranges only over the positive odd integers. The final fraction equals £+3z3-f-5x5-f- — From the coefficient of xn in the expansion of the third sum, we conclude that, if n is even,
where d ranges over all the divisors of n.   Let dl range over the odd values of 6, and 52 over the even values of 6; then
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,
A. Cayley30 discussed the solution for AT of c/>(AO = -AT'.   Set AT = aab* . . . , where a, 6, ... are distinct primes.   Multiply
by the analogous series in b, 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 Nf will be a solution N of 4>(N) =Nf. We need use only the primes a for which a — I divides N', and continue each series only so far as it gives a divisor of Nr for the coefficient of a""1 (a — 1).
V. A. Lebesgue31 proved <t>(Z}= n<£(z) as had Crelle17 and then <f>(z) =H(pi— 1) by the usual method of excluding multiples of pi, . . . , pn in turn. By the last method he proved (pp. 125-8) Legendre's (5), and the more general formula preceding (5).
J. J. Sylvester32 proved (4) by the method of Ettingshausen,7 using (2) instead of (3). By means of (4) he gave a simple proof of the first formula of Dirichlet;21 call the left member vn; since [n/r] — [(n — l)/r] = l or 0, according as n is or is not divisible by r,
The constant c is zero since Ui = l.   He stated the generalization
2f«(i^(r-1+2r--1+. . .+r?lr"1))=:r+2r+. . .+nr.
i«l|^         \                              LfrJ      / J
He remarked that the theorem in its simplest form is
"London Ed. and Dublin Phil. Mag., (4), 14, 1857, 539-540.
"Exercices d'analyse numdriquc, 1859, 43-45.
"Quar. Jour. Math., 3, 1860, 186-190; Coll. Math. Papers, 2, 225-8.