CHAP, xi] MISCELLANEOUS THEOREMS ON DIVISIBILITY. 331 He investigated the related problem of Dirichlet.4 Finally, he used as divisors all the sth powers ^ n and found the ratio of the number of remainders less than half of the corresponding divisors to the number of the others. L. E. Dickson17a and H. S. Vandiver proved that 2n>2(n+l)(n'+l) . . ., if 1, n, n', . . . are the divisors of an odd number n>3. R. Birkeland18 considered the sum sa of the gth powers of the roots Oi, . . ., am of zm+Alzm~l+ . ; . +AW, = 0. If s1} . . . , sm are divisible by the power ap of a prime a, then Aq is divisible by a? unless q is divisible by a. If q is divisible by a, and oP is the highest power of a dividing g, then Afl is divisible by ap~pl. Then (n+aaO . . . (n+aam) -nm is divisible by ap. In particular, the product of m consecutive odd integers is of the form l+2pt if m is divisible by 2P. E. Landau19 reproduced Poussin's16 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 Dirichlet3. Given an infinite class of positive numbers q without a finite limit point and such that the number of g's ^x is asymptotic to x/w(x), where w(x) is a non-decreasing positive function having then if x is divided by all the q's^x, the mean of the fractional parts of the quotients has f or x = oo the limit 1 — C. St. Guzel20 wrote d(ri) for the greatest odd divisor of n and proved in an elementary way the asymptotic formulas for 0 as in Pfeiffer90, Ch. X. A. Axer21 considered the xx'"(71) 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 *>th power, X^2, v^2. Then 2£"i%x>* (n) is given asymptotically by a complicated formula involving the zeta function. F. Rogel22 wrote R*,n for the algebraic sum of the partial remainders t-[t] in (2), with n replaced by z, and obtained where pn is the nth prime and pnx^ z<pn\1. He gave relations between the values of Q\(z) for various z's and treated sums of such values, and tabulated the values of 62(2) and 7^2, n for 2^288. He22a gave many relations 170Amer. Math. Monthly, 10, 1903, 272; 11, 1904, 38-9. 18Archiv Math, og Natur., Kristiania, 26, 1904, No. 10. "Bull. Acad. Roy. Bclgique, 1911, 443-472. "Wiadomosci mat., Warsaw, 14, 1910, 171-180. 21Prace mat. fiz., 22, 1911, 73-99 (Polish), 99-102 (German). Review in Bull, dea sc. math., (2), 38,11, 1914, 11-13. »Sitzungsber. Ak. Wiss. Wien (Math.), 121, Ila, 1912, 2419-52. M«/Md.f 122, Ila, 1913, 669-700. See Rogel"* of Ch. XVIII.