322 History of the Theory of Numbers. [Chap, x 
Taking /(/j, k) = l, we obtain Meissel's" (11), a direct proof of which 
is also given. Taking /(/i, k)=f{h)g{hk), we get 
S S/0>(j/c)=S/(/c)S^(iA:), 
-m- I 
special cases of which yield niany known formulas involving Mobius's func- 
tion ju(n) or Euler's function (f>{n). 
E. Landau^^^ proved the result due to PfeifTer^°, and a theorem more 
effective than that by Piltz^^, having the terms replaced by 0{x°-), where, 
for every e>0, 
k-1 . 
E. Landau^^- extended the theorem of Piltz^^ to an arbitrary algebraic 
domain, defining Tk{n) to be the number of representations of n as the norm 
of a product of k ideals of the domain. 
J. W. L. Glaisher^^, generalizing his^^^ formula, proved that 
Sf[^]^(s)=Sf[^]^(s) + 2g[^]/(s)-F(p)G(p), 
where F(s) =/(!)+ . . . +/(s), G{s)=g{l)+. . . -\-g{s), p = [v^]. A similar 
generalization of another formula by Dirichlet^^ is proved, also analogous 
theorems involving only odd arguments. 
Glaisher^^ 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.^®'^^ 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. Landau^^ gave another proof of the result by Voronoi^^^. He proved 
(p. 2223) that T(n)< 471^/^ 
J. W. L. Glaisher^^^ stated again many of his^^ results, but without 
determining the limits of the errors of the asymptotic formulas. 
S. Minetola^^^ proved that the number of ways a product of m distinct 
primes can be expressed as a product of n factors is 
iy{»"-G)("-')"+(2)(»-2)"--(„:ii)4 
T. H. GronwalP^^ noted that the superior limits for a:= oo of 
aM/x" (a>l), (r{x)/{x\oglogx) 
are the zeta function f (a) and e^, respectively, C being Euler's constant. 
"'Gottingen Nachrichten, 1912, 687-690, 716-731. 
"»Tran8. Amer. Math. Soc, 13, 1912, 1-21. 
»«Quar. Jour. Math., 43, 1912, 123-132. 
^**Ibid., 315-377. Summary in Glaisher.i« 
'"Messenger Math., 42, 1912-13, 1-12. 
i««Il Boll, di Matematica Gior. Sc.-Didat., Roma, 11, 1912, 43-46; cf. Giomale di Mat., 45, 1907, 
344-5; 47, 1909, 173, §1, No. 7. 
"Trans. Amer. Math. Soc, 14, 1913, 113-122. 
