Chap. X] 
Sum and Number of Divisors. 
309 
In particular, if the pi include all the primes in order, we may replace 
N{x) by [x], the greatest integer ^x. Since there are as many divisors 
>a of n as there are divisors <n/a, 
D»+i),(^^^=€+n(a,+l), 
where € = 1 or according as n is or is not a divisor of p. These two formulas 
serve as recursion formulas for the computation of N{n). For the case of 
two primes pi = 2, p2 = 3, 
The functions L satisfy similar formulas and are computed similarly. 
J. W. L. Glaisher^"^ stated a theorem, which reduces for m = l to 
Halphen's,4° 
/S=o-^(n)-3o-^(n-l)+5o-^(n-3)-7(r„(n-6)+9(7^(n-10)-. . . 
= 2s(^'^^{cr^_,(n-l)-(l^+2V^_,(n-3)+(lH2H3V^_,(n-6)-. . . j 
provided m is odd, where k ranges over the even numbers 2, 4, . . ., ?n — 1, 
while 6 = or 6 = 1 according as n is not or is of the form g'(^ + l)/2. As in 
Glaisher^^ for m = l, the series are stopped before any term (Ti{n — n) is 
reached; but, if we retain such terms, we must set 6 = for every n and 
define o-i(O) by 
m+2 
m 
©''^^'^^l^-^'+i^^'K 
<r(0) = 
©.;(o)= 
m+2 
B. 
where Bi, B2, . . . are the Bernoullian numbers. 
Glaisher^°^ stated the simpler generalization of Halphen'*": 
'S+S 2Hk^(!k) f^-'^^'') -3^+V^_,(n-l)+5^+V_,(n-3) - . . . } 
where the summation index k ranges over the even numbers 2, 4, ... , m — 1, 
and m is odd. If we include the terms <T2,_i(0) = ( — !)' 5r/(4r) in the left 
member, the right member is to be replaced by 
5{-iy 
2'"+2(m+2) 
"^Messenger Math., 20, 1890-1, 129-135. 
^o^Ibid., 177-181. 
