CHAP. X] SUM AND NlJMBEE OF DlVISORS. 315 numbers between <$>(m) and a, including the limits, are the numbers x from 1 to a (or those ^a) each taken £= [$(x)/x] times, and the numbers within the limits which are multiples of x are x} 2x,..., £r. For example, if a = 3, <j(>(ra)=900/w2, then <3>(z) = 30/Vz and it is a question of the divisors of 3..., 17; for z=3, £ = 5 and 3 is a divisor of 3, 6, 9, 12, 15. For $(z) =n, a = l, the theorem states that among the divisors of 1,..., n any one x occurs [n/x] tunes and that these divisors are l,...,n;l,..., [n/2]; 1,..., [n/3]; etc. Hence the sum of the divisors of 1,..., n is and their product is He proved (pp. 244-6) that the number of divisors =r (mod s) of 1, 2,..., n equals A+B, where A is the number of integers [n/x] for z = l,..., n which have one of the residues r, r+1,...,$ — 1 (mod s), and B is the number of all divisors of 1,2,..., [n/s]. The number of the divisors 6 of m, such that n and such that 5" divides m/5, 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. Bachmann128 gave an exposition of the work of Dirichlet,14'17 Mer-tens,37 Hermite,57 Lipschitz,58 Cesaro,60 Gegenbauer,77 Busche,123' 127 Schroder.124- 126 N. V. Bougaief129 stated that where d ranges over the divisors > 1 of n, and v = [ Vn] ; where d ranges over the divisors of n for which d2<n. If 6 is any function, where, on the left, d! ranges over all the divisors of n; on the right, only over those ^ [ri*/j]. For 0(d) s 1 , this gives 128Die Analytische Zahlentheorie, 1894, 401-422, 431-6, 490-3. "•Comptes Rendus Paris, 120, 1895, 432-4. He used £ (a, 6), fc(a, 6) with the same meaning as x(&, a), X(b, a) of Lerch,m and £i(n) for <r(n).