CHAP.X] SUM AND NUMBER OP DIVISORS. 285 where d ranges over the divisors of m. He proved (p. 411) that S(-l)m/^=2<r(m/2) -<r(m). J. Liouville25 stated without proof the following formulas, in which d ranges over all the divisors of m, while d=m/d: d) = [r(m) }2, where <t>(d) is the number of integers < d and prime to d, 6(d) is the number of decompositions of d into two relatively prime factors, and the accent on 2 denotes that the summation extends only over the square divisors D2 of m. He gave (p. 184) the latter being implied hi a result due to Dirichlet.15 Liouville26 gave the formulas, numbered (a),. . ., (k) by him, in which X(m) = +1 or —1, according as the total number of equal or distinct prime factors of m is even or odd: = 1 or 0, SX(d00(d)r(5) = 1 or 0, according as m is or is not a square; = 0, The number of square divisors D2 of m is SX(d)r(6). Liouville27 gave the formulas, numbered I-XVIII by him: 16 Jour. deMath6matiquea, (2), 2, 1857, 141-4. "Sur quelques fonctions nurn^riques," 1st article. Here Sabc denotes S(abc). MJ6id., 244-8, second article of his series. 27/6wJ., 377-384, third article of his series.