I SUM AND NUMBER OF DIVISORS. 299 where B(x) and JB~B(w), A =£(?+!); also that S JD(ap)» S x-p+l x»JD+l where and D=jD(n), J0=.D(p+l). It is stated that special cases of these two formulas (here reported with greater compactness) were given by Dirichlet, Zeller, Berger and Cesaro. In the second, take r = l, p = 0, and choose the integers a, ($, b, n so that whence D = 0. If Xr is the number of divisors of r which are of the form bx'+P, we get Change n to TI+! and set /3 = 0, 6=<r = l, whence a=n [also set p we get MeisseFs22 formula (11). Other specializations give the last one of the formulas by Lipschitz,68 and where *"=[V^L ^(r) is ^ne number of odd divisors of r, while 2 = 0 or 1 according as [n/v — J] > ^ — 1 or =v—\. L. Gegenbauer73 proved by use of f-functions many formulas involving his72 functions p, / and divisors dt. Among the simplest formulas, special cases of the more general ones, are r(r2), summed for d, c?2, <^4, where to = Vr/rf2- Other special cases are the fourth and sixth formulas of Liouville,29 the first, third and last of Liouville.26 Beginning with p. 414, the formulas involve also "Sitzungsber. Ak. Wien (Math.), 90, II, 1884, 395-459. The functions used are not defined in the paper. For his ^A, ^, w, we write <rj>, r 6, where 0 is the notation of Liouville.26