History Of The Theory Of Numbers - I

CHAP, xixi INVERSION; FUNCTION ju(n). 445
P. Bachmann20 proved that /(n) =2£:f F(kri) implies that
Write X = [x/n]. Taking F(n) =X, nX, $(Z), whence /(n) = !T(Z), ncr(X)> ), respectively, we obtain Lipschitz's50 (Ch. X) formulas:
Let F(ri) be zero if n is not a divisor of P and write \f/(P/n) for F(ri). Hence if d divides P, f(d)=^(P/kd) implies ^(P)=S/i(d)/(d), where A; ranges over the divisors of P/d, and d over those of P.
D. von Sterneck21 considered a function /(n) with the properties: (i) /(I) = 1 ; (ii) the g. c. d. of f(m) and/(n) isf(d) if d is the g. c. d. of m and n; (iii) for primes p, other than specified ones, one of the numbers /(p=»=l) is divisible by p] (iv) the g. c. d. of /(/>n)//(n) and/(n) divides p. Then if L(ri) is the 1. c. m. of the values of /for all the divisors <n of n, F(n) =f(n) ~r-L(ri) is an integer which can be given the form
The four properties hold for the function defined by the recursion formula /(n) =a/(n — l)-f/3/(n— 2), where a and /3 are relatively prime, with the initial conditions /(!) = !, /(2) = a. For a = 2z, /3 = 6 — a^, we have22
f(n) = ^+Vb)n~(a;"-V&)n.
2\/b
The case a=/3 = l was discussed by Lucas28 of Ch. XVII, and his test for primality holds for the present generalization. The four properties hold also for
if a, b are relatively prime ;23 then f(p — 1) is divisible by p if p is a prime not dividing a, 6 or a— b.
K. Zsigmondy24 gave a generalized inversion formula. Let N be any multiple of the relatively prime integers rii, . . . , nt. Set
where d ranges over those divisors ^m of N which are products of powers
20Die Analytische Zahlentheorie, 1894, 310. "Monatshefte Math. Phys., 7, 1896, 37, 342. "Dirichlet, Werke, 1, 47-62. See Dirichlet," Ch. XVII. 23Zsigmondy, Monatshefte Math. Phys., 3, 1892, 265. "Ibid., 7, 1896, 190-3.