Chap. V] EulER's </)-FuNCTION. 123 
I2, I3,... be those divisors of m = p V • • • ^'^ which are given by the expansion 
of the product 
0(m) = (p--p"-i). . .(r-r-i)=?i-Z2+?3-. . .-^v 
It is proved that 
cl>{m, k) = {k, k)-{k, k)+ . . . -{I, k) 
[called Smith's function by Lucas/^ p. 407] is zero if k<m, but equals 
<^(m) if ^ = m. Hence if to the mth column of A^ we add the columns with 
indices I3, I5,... and subtract the columns with indices I2, Z4, . . . , we obtain 
an equal determinant in which the elements of the mth column are zero 
with the exception of the element <^(m). Hence A^=A^_i0(m), so that 
(6) A^=<^(l)0(2)...0(m). 
If we replace the element 5 = {i, j) by any function /(5) of 5, we obtain a 
determinant equal to i^(l) . . .F{m), where 
nm)=/W-2/g)+S/g)-.,.. 
Particular cases are noted. For /(§) = S''', F{m) becomes Jordan's^"" func- 
tion Jkitn). Next, if /(5) is the sum of the kth. powers of the divisors of 5, 
then F{m)='m!'. Finally, if /(6) = 1^+2^+ . . . +5^, it is stated erroneous- 
ly that F{m) is the sum (i>k{'m) of the /bth powers of the integers ^m and 
prime to m. [Smith overlooked the factors o!', oJ^h^, ... in Thacker's^^° first 
expression for <l>k{n), which is otherwise of the desired form F{n). The 
determinant is not equal to 4>k{\) . . .^^kim), as the simple case k = l, w = 2, 
shows.] 
In the main theorem we may replace 1,. . ., m by any set of distinct 
numbers jui, . . . , jU;„ such that every divisor of each ju, is a number of the 
set; the determinant whose element in the ith. row and jth column is/(5), 
where 5 = (jUi, /xy), equals F()Ui) . . .F{fx^. Examples of sets of ^t'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. Dedekind^° proved that, if n be decomposed in every way into a 
product ad, and if e is the g. o,. &. oi a, d, then 
S|.#,(6) = nn(l+^), 
where a ranges over all divisors of n, and p over the prime divisors of n. 
P. Mansion^^ stated that Smith's relation (6) yields a true relation if we 
replace the elements 1,2,. . .of the determinant A^ by any symbols Xi,X2,. . ., 
and replace 0(m) by Xi^—Xi^-\-Xi — .... [But the latter is only another 
form of Smith's F{m) when we write x^ for Smith's /(5), so that the generali- 
zation is the same as Smith's.] 
"Jour, ftir Math., 83, 1877, 288. Cf. H. Weber, Elliptische Functionen, 1891, 244-5; ed. 2, 
1908 (Algebra III), 234-5. 
^Messenger Math., 7, 1877-8, 81-2. 
