Chap. X] SuM AND NuMBER OF DiVISORS. 287 
ible by the prime 4X+1. The condition given by Bouniakowsky^° is neces- 
sary, but not sufficient. Also, 
o-3(m) = S <T{2j-l)a{2m-2j+l) {m odd). 
J. Liouville's series of 18 articles, "Sur quelques formules . . .utiles dans la 
th^orie des nombres," in Jour, de Math., 1858-1865, involve the function 
(r„, but will be reported on in volume II of this History in connection with 
sums of squares. A paper of 1860 by Kronecker will be considered in 
connection with one by Hermite.'^'^ 
C. Traub^^ investigated the number {N; M, t) of divisors T oi N which 
are = t (mod M) , where M is prime to t and N. Let a,h,. . .,lhe the integers 
< M and prime to M ; let them belong modulo M to the respective exponents 
a', h',. . ., V; let m be a common multiple of the latter. Since any prime 
factor of N is of the form Mx+k, where k = a,. . .,1, any T is congruent to 
a^6^. . .Z^=« (mod M), O^A<a',. . ., O^KV. 
Let A',. . ., L' he one of the n sets of exponents satisfying these conditions. 
If P is a primitive mth root of unity, the function 
1/' = ^7-^SP^ e = {A-A')am/a'+ . . .+{L-L')\m/V, 
summed for all sets 0^a<a', . . ., O^X<r, has the property that i^ = l if 
A = A'(mod a'),. . ., L=L'(mod V) simultaneously, while i/' = in all other 
cases. Thus {N] M, t) =SSt/', where one summation refers to the n sets 
mentioned, while the other refers to the various divisors T of N. This 
double sum is simplified. 
[The properties found (pp. 278-294) for the set of residues modulo M 
of the products of powers oi a,. . ., I may be deduced more simply from the 
modern theory of commutative groups.] 
V. Bouniakowsky^^ considered the series 
n=l'c. n=l "' 
By forming the product of xl/ix)""'^ by \{/{x) , he proved that z„, 2 is the number 
No{n)=T{n) of the divisors of n, and Zn,m equals 
where (and below) d ranges over the divisors of n. Also, 
\p(x)\l/{x-l)= 2) ——• 
n=l '«' 
From \l/{xYxl/(x-iy for (i, j) = (2, 1), (2, 2), (1, 2), he derived the first and 
fourth formulas of Liouville's^^ first article and the fourth of his^^ second 
article. He extended these three formulas to sums of powers of the divisors 
^lArchiv Math. Phys., 37, 1861, 277-345. 
32M^m. Ac. Sc. St. P^tersbourg, (7), 4, 1862, No. 2, 35 pp. 
