Chap. XIX] INVERSION) FUNCTION fx{n). 447 
E. B. EUiott^^ of Ch. V gave a generalization of fx{n). 
L. Kronecker^" defined the function p{n, k) of the g. c. d. (n, k) of n, k 
to be 1 if (n, A:) = 1, if (n, A;)>1, and proved for any function /(n, k) of 
(n, k) the identity 
S p{n, k)f{n, k) =X S [x{d)fin, kd), 
k=l d k=\ 
where d ranges over the divisors of n. The left member is thus the sum of 
the values oif{n, k) ior k<n and prime to n. Set 
Fin, rf) = S f(n, kd) , $(n, d) = "s p (^, k)fin, kd) . 
k=i k=i \tt / 
Thus when d ranges over the divisors of n, 
Fin, 1) =S$(n, d), *(n, 1) =2M(d)F(n, d) 
d 
are consequences of each other. The same is true (p. 274) for 
/i(n)=S/(d)^(^), fin) =i:ixid)gid)h{^, 
if girs) =gir)gis). Application is made (p. 335) to mean values. 
E. M. L^meray^^ gave a generalized inversion theorem. Let i/'2(a, b) be 
symmetrical in a, b and such that the function ^3 defined by 
\f/sia, b, c)=\l/2{a,-ip2ib, c)\ 
is symmetrical in a, b, c. Then the function 
\pM, b, c, d) =\l/3[a, b, yPiic, d)} 
will be symmetrical in a, b, c, d and similarly for t^fc(ai, . . . , 0;^) . For example, 
\l/2ia, b)=aVl-\-b^+bVT+c^, ^3 = a&c+SaVl+6Vl+c2. 
Let v=^iy, u) be the solution of y=yp2i'^j ^) for ^- The theorem states that, 
li di,. . .,dk are the divisors of m = p°gV. . , and if Fim) be defined by 
Fim)=Mfid,),...Jidj:)], 
we have inversely /(m) =Q(G, H), where 
G=4.w,.£),.g),... ,.(-!-),...}, 
^=4K?)-(f)'-<^)'-;} 
where ix is the number of combinations of the distinct prime factors p, 
q,... of m taken 0, 2, 4, ... at a time, and v the number taken 1, 3, 5, ... at a 
time. 
L. Gegenbauer^^ defined ^lix) to be +1 if a: is a unit of the field Rii) of 
complex integers or a product of an even number of distinct primes of 
soVorlesungen iiber Zahlentheorie, I, 1901, 246-257. His €„ is /i(n). 
31N0UV. Ann. Math., (4), 1, 1901, 163-7. 
32Verslag. Wiss. Ak. Wetenschappen, Amsterdam, 10, 1901-2, 195-207 (German.) English 
transl. in Proc. Sect. Sc. Ak. Wet., 4, 1902, 169-181. 
