Chap. X] SuM AND NuMBER OF DiVISORS. 321 
even divisor d. In the case of the theorems on the cancellation of actual 
divisors, the results follow at once from the earlier ones. But the recursion 
formulae for o-„ and f „ are new and too numerous to quote. Cancellation 
formulas (pp. 449-467) are proved for the divisors whose complementary- 
divisors are odd, and applied to obtain recursion formulae for the related 
function A/(n) of Glaisher.''^' " 
E. Landau ^^^ proved that log 2 is the superior limit for x= oo of 
log T(x)-log log a;-^log X. 
M. Fekete^^^ employed the determinant RkX obtained by deleting the 
last t rows and last t columns of Sylvester's eliminant of x'^— 1 = and 
a;"-l = 0. Set, for A;^n, 
Then 6„(A;) = 1 or according as k is or is not a divisor of n; while c„(i, k) = l 
if ik = n and i is relatively prime to k, but = in the contrary cases. Thus 
T(n)=S6„(fc), (T{n)=ikK{k), 
k = \ A: = l 
while the number and sum of those divisors d of n, which are relatively prime 
to the complementary divisors n/d, equal, respectively, 
n 1 " 
S Cn{i, k), - i: (i+k) c^{i, k). 
i, k = \ ^ i,k = l 
J. Schroder^" deduced from his^^"* final equation the results 
The final sum equals XIZI i//(s, [s/{r+l)]). 
P. Bachmann^^^ gave an exposition of the work of Euler,^'^ Glaisher,^^' ^^ 
Zeller,66 stern,^^ Glaisher,iio Liouville.^^ 
E. Landau ^^^ proved that the number of positive integers^ a; which have 
exactly n positive integral divisors is asymptotic to 
Aa;^/^^-^^(log log rr)'"-Vlog x, 
where p is the least prime factor of n, and p occurs exactly w times in n, 
while A depends only on n. 
K. Knopp^^" obtained, by enumerations of lattice points, 
n n w w 
i:Mq,k)= i:hik,q)= i:f,iq,k)+i:f2{k,q)-F{w,w), 
k=l k=l k=l k=l 
where q = [n/k] and 
/i(r, k)=k fij, k), h{k, s)=i f{k, j), F(r, s)=i Mr, j). 
y=i y=i i=i 
i66Handbuch. . .Verteilung der Primzahlen, I, 1909, 219-222. 
i^^Math. 6s Phys. Lapok (Math. phys. soc), Budapest, 18, 1909, 349-370. German transl., 
Math. Naturwiss. Berichte aus Ungarn, 26, 1913 (1908), 196-211, 
>"Mitt. Math. Gesell. Hamburg, 4, 1910, 467-470. 
issNiedere Zahlentheorie, II, 1910, 268-273, 284-304, 375. 
i^Annaes So. Acad. Polyt. do Porto, Coimbra, 6, 1911, 129-137. 
""Sitzungsber. Berlin Math. Gesell., 11, 1912, 32-9; with Archiv Math. Phys. 
