Chap. X] SuM AND Number of Divisors. 299 
where 
and B = B{n), A=B{p+l); also that 
2 D{x)= S F{x)+Dn-Ep, 
z=p+l x=D+l 
where 
and D = D(n), E = 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 t = 1, p = 0, and choose the 
integers a, j8, b, n so that 
hn''^-^>a>h{n-iy-\-^, 
whence D = 0. If Xr is the number of divisors of r which are of the form 
bx'+jS, we get 
Change n to n + 1 and set i3 = 0, 6 = o- = l, whence a = n [also set p = [Vn]]; 
we get Meissel's^^ formula (11). Other speciaHzations give the last one 
of the formulas by Lipschitz,^^ and 
where v = [-\/n\, k{r) is the number of odd divisors of r, while ^ = or 1 
according as [n/v — ^]>v — \ or =v — \. 
L. Gegenbauer'^^ proved by use of ^-functions many formulas involving 
his'^^ functions p, / and divisors d<. Among the simplest formulas, special 
cases of the more general ones, are 
2(r,(d)d^=S(r,+x(0rf'=2(r^(0d^+\ Xn\d,) =SX(A), 
mh)tx\d,) =Xfx\h), i:r{h')}x\d,) =Zd{h), 2m'(^)^(0 ='r(r'), 
Xrid^Uh) =e{r), Xi/{d)J,(^^ ^^dMh), 
summed for d, c?2, d^, where h = Vr/dg. Other special cases are the fourth 
and sixth formulas of Liouville,^^ the first, third and last of Liouville.^^ 
Beginning with p. 414, the formulas involve also 
oi,{n)^n'Ti {l + \/vt), n=ilp:\ 
1=1 1=1 
"Sitzungsber. Ak. Wien (Math.), 90, II, 1884, 395-459. The functions used are not defined 
in the paper. For his \pf,, ^, u, we write 0-^, r, e, where e is the notation of Liouville." 
