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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
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
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.