Chap. XI] MISCELLANEOUS ThEOEEMS ON DIVISIBILITY. 331 
He investigated the related problem of Dirichlet.* Finally, he used as 
divisors all the sth powers ^ n and found the ratio of the number of remain- 
ders less than half of the corresponding divisors to the number of the others. 
L. E. Dickson^^" and H. S. Vandiver proved that 2">2(7i+l)(n' + l) . . ., 
if 1, n, n', . • • are the divisors of an odd number n> 3. 
R. Birkeland^^ considered the sum Sg of the qth. powers of the roots 
Oi, . . ., flm of z'^+Aiz"'~^-\- . . . +^m = 0. If Si, . . . , s^ are divisible by the 
power a^ of a prime a, then A^ is divisible by a" unless q is divisible by 
a. If g is divisible by a, and a^' is the highest power of a dividing q, then 
Afi is divisible by a^~^\ Then (n+aai) . . . (n+aa^) —n"" is divisible by a^. 
In particular, the product of m consecutive odd integers is of the form 
1+2^^ if m is divisible by 2". 
E. Landau^^ reproduced Poussin's^^ proof of the final theorem and added 
a simplification. He then proved a theorem which includes as special cases 
the two of Poussin and the final one by Dirichlet^. Given an infinite class 
of positive numbers q without a finite limit point and such that the number 
of g's ^a; is asymptotic to x/w{x), where w{x) is a non-decreasing posi- 
tive function having 
x=oo w{x) 
then if x is divided by all the q's ^ x, the mean of the fractional parts of the 
quotients has for x = «> the limit 1 — C. 
St. GuzeP° wrote 5(n) for the greatest odd divisor of n and proved in 
an elementary way the asymptotic formulas 
[X] rfi \X\ U^\ 
S 5(n) =|-+0(x), S ^^ =f:r+0(l), 
n=l O n=l n 
for as in Pfeiffer^", Ch. X. 
A. Axer^^ considered the x'''''(^) decompositions of n into such a pair 
of factors that always the first factor is not divisible by a Xth power and 
the second factor not by a z^th power, X^2, v'^2. Then S^iix'"'" (n) is 
given asymptotically by a compHcated formula involving the zeta function. 
F. RogeP^ wrote Rx,n for the algebraic sum of the partial remainders 
<— [i] in (2), with n replaced by 2, and obtained 
Qx(2)=2P,,„-|-i2x.n, Px.n= n (l-:;^x)' 
where p„ is the nth prime and Pn''^ 2<p„+i. He gave relations between the 
values of Qx{z) for various 2's and treated sums of such values, and tabu- 
lated the values of ^2(2) and jB2,n for 2^288. He^^" gave many relations 
I'^Amer. Math. Monthly, 10, 1903, 272; 11, 1904, 38-9. 
"Archiv Math, og Natur., Kristiania, 26, 1904, No. 10. 
"Bull. Acad. Roy. Belgique, 1911, 443-472. 
"Wiadomoaci mat., Warsaw, 14, 1910, 171-180. 
"Prace mat. fiz., 22, 1911, 73-99 (Polish), 99-102 (German). Review in Bull, des sc. math., 
(2), 38, II, 1914, 11-13. 
^Sitzungsber. Ak. Wiss. Wien (Math.), 121, Ila, 1912, 2419-52. 
"«/6id., 122, Ila, 1913, 669-700. See RogeP« of Ch. XVIII. 
