Chap. X] SUM AND NUMBER OF DiVISORS. 289 
The sum of the N's is T{n) ^riD^Mn/D^). Hence 
N^^^rm + l 
which is never an integer other than 1 or 2 when n is odd. If it be 2, t(D^) = 3 
requires that D be a prime. Similarly, for Legendre's symbol (2/a), 
is zero if any prime factor 8^± 3 of n occurs to an odd power, but is 11 (a^H- 1) 
if in n each p, is a prime 8Z±1 and each Qi a prime 8Z±3. For n odd, 
Ngh^i/Nsh^s can not be an integer other than 1 or 2; if 2, D is a prime. 
F. Mertens" proved (11). He considered the number v{n) 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(n) 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) =XiJL^{d), where d ranges over the divisors 
of n. Also, 
fc=i k=i \/c / 
He obtained Dirichlet's^'^ expression \l/{n) for this sum, finding for m a limit 
depending on C and n, of the order of magnitude of \/n log^ n. 
E. Catalan^'^" noted that So-(i)o-(i) =80-3(72) where i-{-j = 4n. Also, if i is 
odd, €r{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 4/i+l over the number of divisors 4/^ — 1. 
H. J. S. Smith^^ proved that, if m = pi''ip2"2. . ., 
..W-2..(^)+S.,(^)-. ..=.•- 
For, if P=l+p'+...+p", P' = l+p'+...+p'"-"', then 
c.(m) = P.P. ... , <.. (^) = P/P, . . . , a. (^J = P/P/Pa .... 
and the initial sum equals (Pi — Pi){P2 — P2) ■ . .=m\ 
J. W. L. Glaisher^^ 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. Halphen^° obtained the recursion formula 
(T(n)=3(r(n-l)-5(r(n-3)+. . . -(-l)"(2a;+l)Jn-^^^^|+. . ., 
"Jour, flir Math., 77, 1874, 291-4. 
^'"Recherches sur quelques produits indefinis, M^m. Ac. Roy. Belgique, 40, 1873, 61-191. 
Extract in Nouv. Ann. Math., (2), 13, 1874, 518-523. 
asProc. London Math. Soc, 7, 1875-6, 211. 
'^Messenger Math., 5, 1876, 52. 
^oBuU. Soc. Math. France, 5, 1877, 158. 
