450 HlSTOKY OF THE THEORY Or NUMBERS. [CHAP. XIX n tt= S JU2(l ;0(V^) (mod 2), where 6(ri) is the number of primes a^n. Other special results were cited under 155, Ch. V; 6, Ch. XI; 217, Ch. XVIII. E. Cesaro560 treated 2/(6) in connection with median and asymptotic formulas. Bougaief57 treated numerical integrals, noting formulas like 2^T where ^(n) is the number of prime factors a, &, . . . of n — aab0. . ., S t(d) = S ^(d)+ S ^(aad) = S ^(ad)+ S \f/(d). «» ft d\n d$ <-, Bougaief68 gave a large number of formulas of the type where, on the left, d ranges over all the divisors of m; while, on the right, d ranges over those divisors of m which do not exceed n, [n/2], . . . , respectively. Bougaief59 gave the relation d\n where p ranges over all primes ^ Vn, and ^(m, ri) is the sum of the fcth powers of all divisors ^m of n, so that £0 is their number, and B(t) is the number of primes ^t. L. Gegenbauer60 noted that the preceding result is a case of X=1 X-l where c?x ranges over the divisors ^X of n. Special cases are (a», n), where |p (w, n) is the sum of the pth powers of the divisors ^ w of n. MaGiornale di Mat., 25, 1887, 1-13. "Mat. Sbornik (Math. Soc. Moscow), 14, 1888-90, 169-197; 16, 1891, 169-197 (Russian). B8/6td., 17, 1893-5, 720-59. "Comptes Rendus Paris, 119, 1894, 1259. 80Monatshefte Math. Phys., 6, 1895, 208.