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Full text of "History Of The Theory Of Numbers - I"

CHAP, xixj                    INVERSION; FUNCTION p(n).                             447
E. B. Elliott96 of Ch. V gave a generalization of n(ri).
L. Kronecker30 denned the function p(n, k) of the g. c. d. (n, k) of n, k to be 1 if (n, k) = 1, 0 if (n, &)>!, and proved for any function f(n, k) of (n, k) the identity
S p(n, k)f(n, k) =S
where d ranges over the divisors of n.   The left member is thus the sum of the values of f(n, k) for k< n and prime to n.   Set
n/d                                          n/d     /      \
F(n, d) = Z/(n, M),          *(*, d) = S^jJ, A?j/(n, fcd).
Thus when d ranges over the divisors of n,
F(n, 1) -2*(n, d),          *(n, 1) -2Mtf)F(n, d)
d
are consequences of each other.   The same is true (p. 274) for A(n) -2/(d)?,          /(n)
if ^(rs) =g(r)g(s).   Application is made (p. 335) to mean values.
E. M. L^meray31 gave a generalized inversion theorem.   Let \l/2(a, 5) be symmetrical in a, b and such that the function ^3 defined by
is symmetrical in a, 6, c.   Then the function
^4(a, 6, c, d) =1^3 {a, 6, ^2fe <Q} will be symmetrical in a, 6, c, d and similarly for 1/^(^1, . . . , ak) .   For example,
fa(a, 6) =aVl+62+&Vl-fa2,          1^3=a6c+2aVl+62Vl+c2.
Let v~tt(y, u) be the solution of t/=^2(^, v) for ^-   The theorem states that, if di, . . ., dk are the divisors of m=p*(fr*. . . and if F(m) be denned by
we have inversely /(m) s=fi((?, ZT), where
where M is the number of combinations of the distinct prime factors p, q,...ofm taken 0, 2, 4, ... at a time, and v the number taken 1, 3, 5, ... at a time.
L. Gegenbauer32 defined n(x) to be -hi if # is a unit of the field R(i) of complex integers or a product of an even number of distinct primes of
30Vorlesungen (ib<?r Zahlentheorie, I, 1901, 246-257.   His cn is p(n). "Nouv. Ann. Math., (4), 1, 1901, 163-7.
g. Wise. Ak. Wetenschappen, Amsterdam, 10, 1901-2, 195-207 (German.)    English transl. in Proc. Sect. Sc. Ak. Wet., 4, 1902, 169-181