CHAPTER XIX. 
INVERSION OF FUNCTIONS; MOBIUS' FUNCTION fi(n); NUMERICAL 
INTEGRALS AND DERIVATIVES. 
Inversion; Function fx{n). 
A. F. Mobius^ defined the function /.i(n) to be zero if n is divisible by a 
square >1, but to be ( — 1)'^ if n is a product of k distinct primes >1, 
while ^i(l) = 1. He employed the function in the reversion of series: 
Fix) = S '-^ imphes fix) = S fxis) -^' 
s=l " s=l " 
His results were expressed in more general form by Glaisher^^ and cited in 
Chapter X. See also E. Meissel,^ who^ noted that 
R. Dedekind"* proved that, if F{m) =2/(d), where d ranges over all the 
divisors of m, then 
+ . 
(2) /w=.w-z.o+..e)-..(^) 
where the summations extend over all the combinations 1, 2, 3, ... at a time 
of the distinct prime factors a, b,. . ., k oi m. The proof follows from a 
distribution of all the factors of m into two sets S and T. Put all the divisors 
of m into set S] all divisors of m/a into set T, all of m/b into T, etc. ; all divi- 
sors of m/iab) into S, all of m/iac) into S, etc.; all of m/{abc) into T, etc. 
Then, with the exception of m itself, every divisor of m occurs as often in the 
set S as in the set T. In particular, for Euler's 0(m), m= 20(d), whence 
For another example, see Dedekind''^ of Ch. VIII. Similarly, F(w) = 
n/(d) implies 
F(m) UF g) . . . 
/(») = 
H^Mi-:)- 
J. Liouville^ stated simultaneously with Dedekind the inversion theorem 
for sums and made the same appHcation to ({)im). 
Liouville^ stated the theorem for sums as a problem. 
iJour. fiir Math., 9, 1832, 105; Werke, 4, 591. He wrote a„ for /x(n). 
Hbid., 48, 1854, 301-316. 
'Observationes quaedam in theoria numerorum, Berlin, 1850, pp. 3-6. 
*Jour. fiir Math., 54, 1857, pp. 21, 25. 
6Jour. de Math^matiques, (2), 2, 1857, 110-2. 
«Nouv. Ann. Math., 16, 1857, 181-2. 
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