Chap. XIX] 
Inversion; Function fj>{n). 
443 
E. Cesaro^^ proved formulas, quoted in Ch. X, which include (3) as a 
special case. His erroneous evaluation of the mean of /x(n) is cited there. 
Cesaro^^ reproduced the general formula just cited and extended it to 
three pairs of functions: 
S/i(d)i^i(^) =l^f2(d)F,(^ =S/3(rf)/^3(^), 
FM =S/2(rf)/3 (^) , F, =2/3/1, Fs =2/^2. 
where, in each, d ranges over the divisors of n. 
Cesaro^^ noted that, if h(ri)-\-k{n) = 1 and 
H{n)=h{p)Hq)..., Kin)=k(p)kiq)..., 
where p, q,. . . are the prime factors of n, then 
Hin) =i:fjiid)K{d), Kin) =Xfx{d)H{d). 
For h{n)=k{n) = 1/2, then H{n)=K{n) is the reciprocal of the number of 
divisors, without square factors, of n. 
Cesaro^^ treated the inversion of series. Let Q{x) = 1 or 0, according as x 
is or is not in a given set Q, of integers. Let Q{x)Q,{y) =U(xy). Let €i{x) be 
functions such that €„{e^(x)} =€„^(x) for every pair of indices a, j8. Then 
F{x)=Xh{o,)fleM], 
where co ranges over all the numbers of Q,, implies that 
fix)=i:H{c^)F{eM}, 
if the sum Xh{d)H{n/d), for d ranging over the divisors of n, equals 1 or 
according as n = 1 orn> 1. Cf . Mobius\ 
N. V. Bougaief^^ considered the function v{x) with the value log p if x 
is a power of a prime p, the value in all other cases. Then, if d ranges 
over the divisors of n, St'(d) =log n implies 2ju(d) log d= —v{n). 
H. F. Baker^'^ gave a generalization of the inversion formula, the state- 
ment of which will be clearer after the consideration of one of his appli- 
cations of it. Let fli,. . ., a„ be distinct primes and S any set of positive 
integers. For k-^n, let F{ai,. . ., a^) denote the set of all the numbers 
in S which are divisible by each of the primes a^t+i, o,k+2,- • ■, cin, so that 
F(ai, . . . , aj =*S. For A; = 0, write F{0) for F, so that i^(0) consists of the 
numbers of S which are divisible by ai,. . ., a„. Returning to the general 
F{ai,. . ., ttk), we divide it into sub-sets. Those of its numbers which are 
divisible by no one of ai, . . ., a^ form the sub-set /(ai, . . ., %)• Those 
divisible by ai, but by no one of 02, ... , ak, form the sub-set /(a2, Os, ■ • • , dk)- 
i^Mem. soc. roy. sc. de Liege, (2), 10, 1883, No. 6, pp. 26, 47, 56-8. 
I'Giornale di Mat., 23, 1885, 168 (175). 
"/bid., 25, 1887, 14-19. Cf. 1-13 for a type of inversion formulas. 
"AnnaU di Mat., (2), 13, 1885, 339; 14, 1886-7, 141-158. 
isComptes Rendus Paris, 106, 1888, 652-3. Cf. Cesaro, ihid., 1340-3; Cesa,ro,i2 pp. 315-320; 
Bougaief, Mat. Sbornik (Math. Soc. Moscow), 13, 1886-8, 757-77; 14, 1888-90, 1-44, 
169-201; 18, 1896, 1-54; Kronecker3« (p. 276); Berger"'' (pp. 106-115); Gegenbauer^^ of 
Ch. XI— all on -Ln^d) log d. 
"Proc. London Math. Soc, 21, 1889-90, 30-32. 
