'y-'.J 
Chap. X] SuM AND NuMBER OF DiVISORS. 317 
where i = 3, 5, 7,. . . in S', while the a's range over the solutions of 
ai+. . .+ar-3 = i, ai+2a2+. . . +(r-3)a,_3 = r. 
The case n = leads to relations for T{r). 
J. de Vries^^^" proved the first formula of Lereh's.^^'^ 
A. Berger"^ considered the excess \l/{k) of the sum of the odd divisors of k 
over the sum of the even divisors and proved that 
xl/{n)+x{/in-l)+\l/{n-3)+x(/{n-Q)+xl/in-10)+ . . .=0 orn, 
according as n is not or is a triangular number; also Euler's (2). 
J. FraneP^^ employed two arbitrary functions /, g and set 
d{n) ^Xmg( fj, F{n) = ifU), G{n) = S gU), 
where d ranges over the divisors of n. Then 
id{j) = 2 /(r)G[^] +^j(r)F\j] -F{v)G{v), 
where v = ['\^n]. The case f{x)=g(x) = l gives Meissel's^" (11). Next, he 
evaluated St?(j), where ??(n) =Xf{x)g{y)h(z), sunmied for the sets of positive 
integral solutions of xyz = n. In particular, ■d(n) is the number of such sets 
if f=g = h = l. Using Dirichlet's series, it is shown (p. 386) that 
2^(i) = Si(logn+3C-l)2-3C2+6Ci+l}+e, 
where e is of the order of magnitude of nP^^ log n, C is Euler's constant and 
Ci = 0.0728 . . . [Piltz,^2 Landau^"]. 
FraneP^^ proved that 
2^ = 1 log' P+2C log p+e+Ao, 
r=i r 
where Aq is a coefficient in a certain expansion, and ep^^^ remains in absolute 
value inferior to a fixed number for every p. 
E. Landau ^^'^ gave an immediate proof of (11) and of 
S7^3(^)=2T(.)r-l 
y = l u=l Li'J 
where T^iv) is the number of decompositions of v into three factors. He 
obtained by elementary methods a formula yielding the final result of 
R. D. von Sterneck^^^" proved Jacobi's^^ formula for s^. 
i33aK. Akad. Wetenschappen te Amsterdam, Verslagen, 5, 1897, 223. 
i3*Nova Acta Soc. Sc. UpsaUensis, (3), 17, 1898, No. 3, p. 26. 
"^Math. Annalen, 51, 1899, 369-387. 
i3676id., 52, 1899, 536-8. 
i"/6ici., 54, 1901, 592-601. 
i^^Sitzungsber. Ak. Wiss. Wien (Math.), 109, Ila, 1900, 31-33. 
