Chap. X] SuM AND NuMBEK OF DiVISORS. 315 
numbers between <^(m) and a, including the limits, are the numbers x from 
1 to o (or those ^a) each taken ^= [$(x)/x] times, and the numbers within 
the limits which are multiples of x are x,2x,. . ., ^x. For example, if a = 3, 
</)(m)=900/m^, then <I>(a;)=30/Vx and it is a question of the divisors of 
3. . ., 17; for x = 'd, ^ = 5 and 3 is a divisor of 3, 6, 9, 12, 15. For«I>(x)=n, 
a = l, the theorem states that among the divisors of 1,. . ., n any one x 
occurs [n/x] times and that these divisors are l,...,7i;l,..., [n/2]; 1,. . ., 
[w/3]; etc. Hence the sum of the divisors of 1, . . . , n is 
and their product is 
n3.[n/x]=n[nA]!. 
x=l x=\ 
He proved (pp. 244-6) that the number of divisors =r (mod s) of 1, 2, . . . , n 
equals A-\-B, where A is the number of integers [n/x] for x = \,. . ., n 
which have one of the residues r, r+l,...,s — 1 (mod s), and B is the number 
of all divisors of 1, 2, . . . , [n/s\. The number of the divisors b of m, such that 
\ n 
n 
and such that 5" divides m/8, equals the number of divisors of 1, 2, . . ., n. 
The number of primes among n, [n/2], . . ., [n/n] equals the number of those 
divisors of 1, . . ., n which are primes decreased by the number of divisors 
which exceed by unity a prime. 
P. Bachmann^^^ gave an exposition of the work of Dirichlet,^^' ^^ Mer- 
tens,^'^ Hermite,^^ Lipschitz,^^ Ces^ro,^'' Gegenbauer,'^'^ Busche,^^^' ^" 
Schr6der.i24. 126 
N. V. Bougaiefi29 stated that 
where d ranges over the divisors > 1 of n, and v = [Vn] ; 
-i^HMM' 
where d ranges over the divisors of n for which d <n. If 6 is any function, 
nZ-^d{d)= Xi:d{d), 
a y=i d 
where, on the left, d ranges over all the divisors of n; on the right, only over 
those ^ [nVj]. For 6{d) = l, this gives 
„a(n) = Sx(n,[y]). 
"8Die Analytische Zahlentheorie, 1894, 401-422, 431-6, 490-3. 
i^'Comptes Rendus Paris, 120, 1895, 432-4. He used $ (a, 6), ^(a, b) with the same meaning 
as xib, a), X{b, a) of Lerch,"^ and fi(n) for (r{n). 
