CHAP. X] SUM AND NUMBER OF DIVISORS. 289
The sum of the N's is r(n) =r(D2)r(n/D2). Hence
which is never an integer other than 1 or 2 when n is odd. If it be 2, r (D2) = 3 requires that D be a prime. Similarly, for Legendre's symbol (2/a),
is zero if any prime factor 8Z=t 3 of n occurs to an odd power, but is II (ct;+ 1) if in n each p; is a prime 8Z=*=1 and each & a prime 8Z±3. For n odd, N8h*i/N8h*3 can not be an integer other than 1 or 2; if 2, D is a prime.
F. Mertens37 proved (11). He considered the number v(ri) of divisors of n which are not divisible by a square >1. Evidently v(n) =2", where p is the number of distinct prime factors of n. If ju(tt) is zero when n has a square factor > 1 and is + 1 or — 1 according as n is a product of an even or odd number of distinct primes, v(n) =S//(d), where d ranges over the divisors of n. Also,
2K*)-2M(*)r(pY «-[VnJ.
fc-i A=I VG /
He obtained Dirichlet's17 expression $(ri) for this sum, finding for m a limit depending on C and n, of the order of magnitude of \/n loge n.
E. Catalanf7" noted that Scr(t)or(j) =80*3(71) where i+j=4n. Also, if i is odd, o-(i) equals the sum of the products two at a time of the E's of the odd numbers whose sum is 2i, where E denotes the excess of the number of divisors 4ju+l over the number of divisors 4jut — 1.
H. J. S. Smith38 proved that, if 7n = p1aip2a2- - -,
For, if P = l+p'+ . . . +p", P' = l+p'+ . . . +p<s-1)a, then
P.'ft. . ., ,.- =P1'P2'P3.
and the initial sum equals (Pi— PIf)(Pz — P%). .. =ra*.
J. W. L. Glaisher39 stated that the excess of the sum of the reciprocals of the odd divisors of a number over that for the even divisors is equal to the sum of the reciprocals of the divisors whose complementary divisors are odd. The excess of the sum of the divisors whose complementary divisors are odd over that when they are even equals the sum of the odd divisors.
G. Halphen40 obtained the recursion formula
o-(n)=3o-(n-l)-5<r(n-3) + . . .-(-I*"'0- ' "~f~ X(X±1Y(
37Jour. fur Math., 77, 1874, 291-4.
37aRecherches sur quclques produits ind^finis, M<5m. Ac. Roy. Belgique, 40, 1873, 61-191.
Extract in Nouv. Ann. Math., (2), 13, 1874, 518-523. a'Proc. London Math. Soc., 7, 1875-6, 211. "Messenger Math., 5, 1876, 52. "Bull. Soc. Math. France, 5, 1877, 158.