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CHAP. X]                         SUM AND NuMBEK OF DIVISORS.                               287
ible by the prime 4X+1.   The condition given by Bouniakowsky20 is necessary, but not sufficient.   Also,
er3(m) = 5 <r(2j-l)ff(2m-2j+l)          (m odd).
J. Liouville's series of 18 articles, "Sur quelques formules . . .utiles dans la the*orie des nombres," in Jour, de Math., 1858-1865, involve the function <rn, 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.70
C. Traub31 investigated the number (JV; M, t) of divisors T of N which are = t (mod M) , where M is prime to t and N. Let a, 6, . . . , I be the integers < M and prime to M ; let them belong modulo M to the respective exponents a', b'j . . . , 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, . . ., I, any T is congruent to
aAb*. . .lL=t (mod M),          QA<a',. . ., O^L<!'.
Let A', . . . , Lr be one of the n sets of exponents satisfying these conditions. If P is a primitive rath root of unity, the function
*          e = (A-A')am/a'+...+(L-L')\m/r,
summed for all sets Oga<a', . . ., O^X<T, has the property that ^ = 1 if A=A'(mod a').,. . , I/=Z/(mod V) simultaneously, while ^ = 0 in all other cases. Thus (Nm,-M, t) =S2^, 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 of a, . . . , I may be deduced more simply from the modern theory of commutative groups.]
V. Bouniakowsky32 considered the series
By forming the product of \(/(x)m"~l by \t/(x) , he proved that zn> 2 is the number N0(n) =r(n) of the divisors of n, and zn>m equals
where (and below) d ranges over the divisors of n.    Also,
From iKaOV(s-iy for (i, j) =.(2, 1), (2, 2), (1, 2), he derived the first and fourth formulas of Liouville's25 first article and the fourth of his26 second article. He extended these three formulas to sums of powers of the divisors
31Archiv Math. Phys., 37, 1861, 277-345.
32M<5m. Ac. Sc. St. Pgtersbourg, (7), 4, 1862, No. 2, 35 pp.