Chap. X] SuM AND NuMBER OF DiVISORS. 305 
sum t/'fc'(a;) of the kth powers of the divisors > \/x of x over the sum of the 
A;th powers of the remaining divisors, it follows at once that 
Also 
n V ['fi~\ 
x=l x=lL-CJ 
^J,\x) =£'S^([3) +^''+M - (^+i)5,(.), 
with a similar formula for ^^^(a;), where "^k(^) is the excess of xpki^) over the 
sum of the A;th powers of the divisors < y/x of x. For k — l, the last formula 
reduces to the third one of Hermite's. 
Let Xk{^) be the sum of the kth powers of the odd divisors of x; Xk\^) 
that for the odd divisors > \/x; Xk"{x) the excess of the latter sum over the 
sum of the A;th powers of the odd divisors < ^/x of x; Xk"'{'^) the excess of the 
sum of the kih. powers of the divisors 8s±l>\/x of x over the sum of the 
kih. powers of the divisors 8s='=3<\/^ of ^- For y = 2x and y = 2x — l, the 
sum from x = l to a: = n of Xkiv), Xk'iv), X/iy) and Xk"{y) are expressed as 
complicated sums involving the functions Sk and [x\. 
E. Pfeiffer^° attempted to prove a formula like (7) of Dirichlet/^ where 
now e is 0{'n}^^^^) for every k>0. Here Og{T) means a function whose 
quotient hy g{T) remains numerically less than a fixed finite value for all 
sufficiently large values of 7". E. Landau^ ^ noted that the final step in 
the proof fails from lack of uniform convergence and reconstructed the 
proof to secure this convergence. 
L. Gegenbauer,^^ in continuation of his^° paper, gave similar but longer 
expressions for 
S r{y), S (Tkiy) (2/ = 4a;+l, 6a;+l, 8a:+3, 8a:+5, 8a:+7) 
and deduced similar tests for the primality of y. 
Gegenbauer^^" found the mean of the number of divisors \x-\-a of a 
number of s digits with a complementary divisor iiy-\-^; also for divisors 
ax'^+hy'^. 
Gegenbauer^^'' evaluated A(l) + . . . +A(n) where A{x) is the sum of the 
pth powers of the crth roots of those divisors d oi x which are exact o-th 
powers and whose complementary divisors exceed kdJ/'^. A special case 
gives (11), p. 284 above. 
Gegenbauer^^" gave a formula involving the sum of the A;th powers of 
those divisors of 1, . . . , m whose complementary divisors are divisible by no 
rth power >1. 
'"Ueber die Periodicitat in der Teilbarkeit . . . , Jahresbericht der Pf eiffer'schen Lehr- und Erzieh- 
ungs-Anstalt zu Jena, 1885-6, 1-21. 
"Sitzungsber. Ak. Wiss. Wien (Math.), 121, Ila, 1912, 2195-2332; 124, Ila, 1915, 469-550. 
Landau.^^^ 
»276id., 93, II, 1886, 447-454. 
"-^Sitzungsber. Ak. Wiss. Wien (Math.), 93, 1886, II, 90-105. 
92676id., 94, 1886, II, 35-40. 
"^c/feid., 757-762. 
