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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)(xri)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 -  , PI    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.
6Annales 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).