450 History of the Theory of Numbers. [Chap, xix 
u = l u = l L"J 
ld(^fj^e(x^ (mod 2), 20^^) -2d(£j^d{^n) (mod 3), 
where 6{n) is the number of primes a^n. Other special results were cited 
under 155, Ch. V; 6, Ch. XI; 217, Ch. XVIIL 
E. Ces^ro^^" treated 2/(5) in connection with median and asymptotic 
formulas. 
Bougaief^^ treated numerical integrals, noting formulas like 
liP'"' 
where ^(n) is the number of prime factors a, b,. . . of n = a°6^ . . . , 
2 rPid) = 2 \l/(d)+ 2 rPia'd) = 2 yp{ad)+ 2 yp{d). 
'^i" d\i dr d\i dL^ 
Bougaief^^ gave a large number of formulas of the type 
2V^(d)[^] =2V(rf)+2"*/^V(^)+Sf'»/^V(rf)+ • • ., 
where, on the left, d ranges over all the di\isors of m; while, on the right, d 
ranges over those divisors of m which do not exceed n, [n/2], . . ., respectively. 
Bougaief°^ gave the relation 
Xd(Vd)=^U^2'n): 
d\n P \P / 
where p ranges over all primes ^ -y/n, and ^^(t??, n) is the sum of the kth. 
powers of all di\'isors ^m of n, so that ^o is their number, and ^(0 is the 
number of primes ^t. 
L. Gegenbauer^'^ noted that the preceding result is a case of 
S,(d).(JJ/(X)=J/(X){2,W.(3}, 
where dx ranges over the divisors ^X of n. Special cases are 
Y%(<f) =yUa'. n), ^d%{(^)^^2a%{l, n), 
where |p {m, n) is the sum of the pth powers of the di\'isors ^m of n. 
«^Giornale di Mat., 2.5, 18S7, 1-13. 
"Mat. Sbornik (Math. Soc. Moscow), 14, 1888-90, 169-197; 16, 1891, 169-197 (Russian). 
"Ibid., 17, 1893-5, 720-59. 
"Comptes Rendus Paris, 119, 1894, 1259. 
•oMonatshefte Math. Phys., 6, 1895, 208. 
