CHAP. X] SUM AND NUMBEH OF DIVISORS. 307 where t ranges over the odd integers ^m. For the K and I/ of Lipschitz84 and G(m) =cr(l) -f<r(2) + . . . +<r(m), it is shown that (mod 2), (mod 2). J. Hacks97 gave a geometrical proof of (11) and of Dirichlet's17 expression for T(ri), just preceding (7). He proved that the sum of all the divisors, which are exact ath powers, of 1, 2, . . . , m is He gave (pp. 13-15) several expressions for his96 3f(m), ®(m), K(m). L. Gegenbauer970 gave simple proofs of the congruences of Hacks.96 M. Lerch98 considered the number \l/(at b) of divisors >b of a and proved that [n/2] n (15) S iKn-p,p)=n, A. Strnad" considered the same formulas (15). M. Lerch100 considered the number x(a> b) of the divisors ^fe of a and proved that {^(m—cra, fc+a-)— xC^"^^ a)} , a)}=0. This reduces to his (15) for a = l, fc = l orm+1. Let (Jfc, n; w) denote the g. c. d. (k, n) of /k, n or zero, according as (A;, n) is or is not a divisor of m. Then (16) S {^(m+an, a)— ^(m+an, a)} = 2J (A;, n; m). a=0 fc-1 In case m and n are relatively prime, the right member equals the number 0(a, n) of integers^ a which are prime to n. Finally, it is stated that (17) i if/(m-an, a) = S xfai-exn, n), c= f^— a»o a-o L n Gegenbauer,29 Ch. VIII, proved (16) and the formula preceding it. 97Acta Math., 10, 1887, 9-11. 97a8itzungsber. Ak. Wiss. Wien (Math.), 95, 1887, II, 297-8. 98Prag Sitzungsberichte (Math.), 1887, 683-8. "Casopis mat. fys., 18, 1888, 204. "°Compt. Rend. Paris, 106, 1888, 186. Bull, des ac. math, et astr., (2), 12, 1, 1888, 100-108, 121-6.