CHAP. X] SUM AND NUMBER OP DIVISORS. 319 H. Mellin144 obtained asymptotic expressions for £r(n), 2<r(n). I. Giulini145 noted that, if m and h are given integers, and /3(r) is the sum of the divisors d=mk+h of r, then d], t-0, 1,. . ., [(n-ft)/ro]. The number and sum of the divisors d=mk+h of 1, . . . , n are mr / ««iL ms respectively, where £72(^) = [x][a?+l]/2. G. Vorono'i145a gave for T(x) the precise analytic expression -2 C J* g(t)dt+ and (p. 515) approximations to these integrals, where •w - -* - He discussed at length the function g(x) and (pp. 467, 480-514) the asymptotic value of 2r(ri)(x—ri)k/kl. J. Schroder146 proved that the sum of the pth powers of 1, . . . , n is n r>n n~1 ' rni S f - =n<r,_1(W)+ S p'+S'p« - , P»I LpJ P-<-J-I P»=i LpJ where <=[n/2], and the accent on the last S denotes that the summation extends only over the values 5* t of p which are not divisors of n. E. Busche147 proved that, if we multiply each divisor of m by each divisor of n, the number of times we obtain a given divisor a of mn is r(jLo>/a), where M is the g. c. d. of m,a, and v is that of n, a. A like theorem is proved for the divisors of mnp .... He stated (p. 233; cf. Bachmann168) that where d ranges over the common divisors of m, n. C. Hansen148 denoted by Ti(ri) and T3(n) the number of divisors of n of the respective forms 4/c — 1 and 4fc — 3, and set By use of Jacobi's 03(v, s) for v = 1/4, he proved that 2 A s4"-3- 2 f IV** "s -(~1) 144Acta Math., 28, 1904, 49. 146Giornale di mat., 42, 1904, 103-8. »6«Annales sc. 1'Scole norm, sup., (3), 21, 1904, 213-6, 245-9, 258-267, 472-480. Cf . Hardy.180 148Mitt. Math. Gesell. Hamburg, 4, 1906, 256-8. "7/6i<i, 4, 1906, 229. U80versigt K. Danske Videnskabernes Selskabs Forhandlinger, 1906, 19-30 (in French).