Chap. X] SuM AND NUMBER OF DiVISORS. 319 
H. Mellin"^ obtained asymptotic expressions for2T(n), So-(n). 
I. Giulini^^^ noted that, if m and h are given integers, and /3(r) is the sum 
of the divisors d = mk-\-h of r, then 
i8(l) + . . .+/3(n)=2:4n/d], A; = 0, 1,. . ., [{n-h)/m], 
k 
The number and sum of the divisors d='mk-\-h oil,. . ., n are 
[(n-w«]r^-i [n/im+h)] /^-hA , ,%'^\-n-sh , ,-| 
S hj ' ^ 2 Eal ]+hi:\ +1 h 
fc=o LttJ r=i \ mr / s=iL ms J 
respectively, where ^2(2^) = W[a^+l]/2. 
G. Voronoi^^^" gave for T{x) the precise analytic expression 
x(logx+2(7-l)+i+Ma^)-2| g{t)dt^ {g{-x+ti)-g(-x-U)}idt, 
and (p. 515) approximations to these integrals, where 
,(x)= -i log .-iC-!2i^+2^ i^x(») ,og5(^+-i-). 
He discussed at length the function g{x) and (pp. 467, 480-514) the asymp- 
totic value of Sr(n)(x — 7i)V^!- 
J. Schroder^^® proved that the sum of the I'th powers of 1, . . . , n is 
S pf-1 =n(T._i(n)+ "S p'+ r pT ^1, 
p=i LpJ p=t+i P=i LPJ 
where t = [n/2], and the accent on the last S denotes that the summation 
extends only over the values ^ ^ of p which are not divisors of n. 
E. Busche^*^ 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 Tiixv/a), where 
jjL is the g. c. d. of m,a, and v is that of n, a. A like theorem is proved for 
th^ divisors of mnp .... He stated (p. 233; cf. Bachmann^^^) that 
(Th{m)(Th{n) =SdV;,f ^j, 
where d ranges over the common divisors of m, n. 
C. Hansen ^^^ denoted by Ti{n) and T^{n) the number of divisors of n 
of the respective forms 4/c — 1 and 4/^ — 3, and set 
A„=r3(4n-3)-ri(4n-3). 
By use of Jacobi's B^{v, s) for ^ = 1/4, he proved that 
„=i '^^ ~Zx^ ^ l-s'^"-^ l-2s^+2s^«+... 
i«Acta Math., 28, 1904, 49. 
i«Giornale di mat., 42, 1904, 103-8. 
i^'^Annales sc. I'ecole norm, sup., (3), 21, 1904, 213-6, 245-9, 258-267, 472-480. Cf. Hardy.i^" 
"«Mitt. Math. GeseU. Hamburg, 4, 1906, 256-8. 
"V6id., 4, 1906, 229. 
"^Oversigt K. Danske Videnskabemes Selskabs Forhandlinger, 1906, 19-30 (in French). 
