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CHAP. X]                       SUM AND NUMBEK 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 <rn and f n 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.74' 87
E. Landau155 proved that log 2 is the superior limit for a; =00 of log T (x) -log log x-r-log x.
M. Fekete166 employed the determinant R^n obtained by deleting the last t rows and last t columns of Sylvester's eHminant of #*— 1 = 0 and zn-l=0. Set, forfcgn,
wjo-i-ififc'1'!, cn(a)Hflai(iHflOT^
Then bn(k) = 1 or 0 according as k is or is not a divisor of n; while cn(i, fc) = 1 if ik=n and i is relatively prime to k, but =0 in the contrary cases. Thus
r(n) = 2 &»(*),          <r(n)» 2 »,(*),
jfc-i                                jfc-i
while the number and sum of those divisors d of n, which are relatively prime to the complementary divisors n/d, equal, respectively,
2 cn(i, k),          \ 2 (i+k) cn(iy t).
»'.*-l                        A',*-l
J. Schroder157 deduced from his124 final equation the results
The final sum equals S;:j ^(s, [a/(r+l)]).
P. Bachmann158 gave an exposition of the work of Euler,5'6 Glaisher,63'64 Zeller,66 Stern,85 Glaisher,110 Liouville.30
E. Landau159 proved that the number of positive integers ^x which have exactly n positive integral divisors is asymptotic to
Ax^-^log log x)»-l/log x,
where p is the least prime factor of n, and p occurs exactly w times in nt while A depends only on n.
K. Knopp160 obtained, by enumerations of lattice points,
where q — [n/k] and
/!(r, fc) = 2 f(j, *),          /a(fc, s) = 2 /(*, j),          F(r, s) = 2 A(r, j).
_ y-i _ y-i _ y-i
166Handbuch. . .Verteilung der Primzahlen, I, 1909, 219-222.
"•Math. 6s Phys. Lapok (Math. phys. soc.), Budapest, 18, 1909, 349-370.    German trans
Math. Naturwiss. Berichte aua Ungarn, 26, 1913 (1908), 196-211. "7Mitt. Math. GeseU. Hamburg, 4, 1910, 467-470.                                         *
»"Niedere Zahlentheorie, II, 1910, 268-273, 284-304, 375.