History Of The Theory Of Numbers - I

CHAP. XIX] INVERSION; FUNCTION /*(n). 443
E. Cesaro12 proved formulas, quoted in Ch. X, which include (3) as a special case. His erroneous evaluation of the mean of n(ri) is cited there.
Cesaro13 reproduced the general formula just cited and extended it to three pairs of functions:
Fi(n) =S/2 (d)/8 ~
where, in each, d ranges over the divisors of n. Cesaro14 noted that, if h(ri)+k(ri) = 1 and
where p, g, . . . are the prime factors of n, then H(n) ~%n(d)K(d), K(ri)
For h(ri) =k(ri) = 1/2, then ff(n) =X(n) is the reciprocal of the number of divisors, without square factors, of n.
Cesaro18 treated the inversion of series. Let fi(x) = 1 or 0, according as x is or is not in a given set Q of integers. Let Q(oOd(t/) ~ti(xy). Let et(aO be functions such that ea{€ft(x)} =€^(x) for every pair of indices a, 0. Then
where w ranges over all the numbers of ft, implies that
if the sum %h(d)H(n/d), for d ranging over the divisors of n, equals 1 or 0 according as n = 1 or n> 1. Cf . Mobius1.
N. V. Bougaief16 considered the function v(x) with the value log p if x is a power of a prime p, the value 0 in all other cases. Then, if d ranges over the divisors of n, 2*>(d) =log n implies Sju(d) log d= — v(n).
H. F. Baker17 gave a generalization of the inversion formula, the statement of which will be clearer after the consideration of one of his applications of it. Let <*!,..., an be distinct primes and $ any set of positive integers. For k^n, let F(alj. . ., a*) denote the set of all the numbers in S which are divisible by each of the primes a^i, ak+2, . . . , anj so that F(alt . . . , an) =S. For k = 0, write F(Q) for F} so that F(0) consists of the numbers of S which are divisible by ax, . . . , an. Returning to the general F(ai,. . ., a*), we divide it into sub-sets. Those of its numbers which are divisible by no one of alf. . ., ah form the sub-set f(a1}. . ., a*). Those divisible by alt but by no one of 02, . . ., ak) form the sub-set /(a2, a3, . . ., a*).
12M6m. soc. roy. sc. de Li&ge, (2), 10, 1883, No. 6, pp. 26, 47, 56-8.
"Giornaledi Mat., 23, 1885, 168 (175).
"Ibid., 25, 1887, 14-19. Cf. 1-13 for a type of inversion formulas.
"Annali di Mat., (2), 13, 1885, 339; 14, 1886-7, 141-158.
"Comptes Rendua Paris, 106, 1888, 652-3. Cf. Cesaro, ibid., 1340-3; Cesaro," 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; Kronecker30 (p. 276); Berger17" (pp. 106-115); Gegenbauer12 of Ch. XI— all on SM(cZ) log d.
17Proc. London Math. Soc., 21, 1889-90, 30-32.