History Of The Theory Of Numbers - I

CHAP.V] EULER'S ^-FtiNCTioN. 123
12) Z3, . . . be those divisors of m=p*qf . . .tr which are given by the expansion of the product
<t>(m) = (p'-p--1) . . . (r-
It is proved that
*(m,A;)s(Z1,fc)-(Z
[called Smith's function by Lucas,72 p. 407] is zero if k<m, but equals 0(m) if fc = m. Hence if to the mth column of Am we add the columns with indices Z3, 15, . . . and subtract the columns with indices Z2,'J4, . . ., we obtain an equal determinant in which the elements of the mth column are zero with the exception of the element <£(w). Hence Am=Am_!0(m), so that
(6) Am=0(l)(/>(2)...0(m).
If we replace the element 5 = (i, j) by any function /(S) of d, we obtain a determinant equal to F(l) . . .F(m), where
Particular cases are noted. For jf(6)=5fc, F(m) becomes Jordan's200 function Jk(m). Next, if /(5) is the sum of the kth powers of the divisors of 5, then F(m) =mk. Finally, if /(5) = lfc+2fc+ . . . +5*, it is stated erroneously that F(m) is the sum ^(w) of the /cth powers of the integers ^ra and prime to w. [Smith overlooked the factors ak, akbk, ... in Thacker's150 first expression for <t>k(n)t which is otherwise of the desired form F(n). The determinant is not equal to <£*(!) . . .<fo.(m), as the simple case k = 1, ra = 2, shows.]
In the main theorem we may replace 1,. . ., m by any set of distinct numbers /zi, . . . , jiim such that every divisor of each ^ is a number of the set; the determinant whose element in the ith row and jth column is /(5), where 5= (/z{, ^-), equals ^(^0 . . .F(nm). Examples of sets of M'S are the numbers in their natural order with the multiples of given primes rejected; the numbers composed of given primes; and the numbers without square factors.
R. Dedekind40 proved that, if n be decomposed in every way into a product ad, and if e is the g. c. d. of a, d, then
PS where a ranges over all divisors of n, and p over the prime divisors of n.
P. Mansion41 stated that Smith's relation (G) yields a true relation if we replace the elerrients 1,2,. . .of the determinant Am by any symbols x-i, x>2,..., and replace <K?tt) by xti—x^+x^— .. .. [But the latter is only another form of Smith's F(m) when we write x5 for Smith's/(6), so that the generalization is the same as Smith's.]
"Jour, fur Math., 83, 1877, 288. Cf. H. Weber, Elliptiache Functional, 1891, 244-5; ed. 2,
1908 (Algebra III), 234-5. "Messenger Math., 7, 1877-8, 81-2.