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322                          HlSTOBY OF THE THEOBY OF NtJMBEBS.                  [CHAP. X
Taking f(h, &)==!, we obtain MeisseFs22 (11), a direct proof of which is also given.   Taking f(h, k)=f(h)g(hk), we get
Z   2f(j)gW~2m2g(jk),        ff-Rl fc»i y-i                   fc»i        y-i                           L/CJ
special cases of which yield many known formulas involving Mobius's function p,(ri) or Euler's function 4>(n).
E. Landau161 proved the result due to Pfeiffer90, and a theorem more effective than that by Piltz52, having the 0 terms replaced by 0(xa), where, for every €>0,
fc — 1
E. Landau162 extended the theorem of Piltz52 to an arbitrary algebraic domain, denning Tk(ri) to be the number of representations of n as the norm of a product of k ideals of the domain.
J. W. L. Glaisher163, generalizing his142 formula, proved that
where F(«) =/(!)+. ..+/(«), <?(«) «?(!)+. . .+0(0, P = [Vn|. A similar generalization of another formula by Dirichlet17 is proved, also analogous theorems involving only odd arguments.
Glaisher164 applied the formulas just mentioned to obtain theorems on the number and sum of powers of divisors, which include all or only the even or only the odd divisors. Among the results are (11) and those of Hacks.96'97 The larger part of the paper relates to asymptotic formulas for the functions mentioned, and the theorems are too numerous to be cited here.
E. Landau91 gave another proof of the result by Voronoi143. He proved (p.2223) thatr(7i)<4n1/3.
J. W. L. Glaisher165 stated again many of his164 results, but without determining the limits of the errors of the asymptotic formulas.
S. Minetola166 proved that the number of ways a product of m distinct primes can be expressed as a product of n factors is
T. H. Gronwall167 noted that the superior limits for x = oo of
(ra(x)/xa    (a>l),          <r(z)/(zloglogz)
are the zeta function f (a) and ec, respectively, C being Euler's constant.
wlGottingen Nachrichten, 1912, 687-690, 716-731. 16*Trans. Amer. Math. Soc., 13, 1912, 1-21. lMQuar. Jour. Math., 43, 1912, 123-132. l"Ibid., 315-377.   Summary in Glaisher.165 "•Messenger M#th., 42, 1912-13, 1-12.
188I1 Boll, di Matematica Gior. Sc.-Didat., Roma, 11, 1912, 43-i6; cf . Giornale di Mat., 45, 1907, 344-5; 47, 1909, 173, §1, No. 7. . Amer. Math. Soc., 14, 1913, 113-122.