Chap. V] GENERALIZATIONS OF EuLER's 0-FuNCTION. 147 
Schemmel's Generalization of Euler's ^-Function. 
V. SchemmeP^° considered the $„(m) sets of n consecutive numbers each 
<m and relatively prime to m. If m = a'^}f . . ,, where a, h,. . . are distinct 
primes, and m, m' are relatively prime, he stated that 
$„(m) =a"~^(a — n)6^~-^(5— n) . . ., <^n{mm') =<l>„(m)$n(m'), 
Sn"-''V-^'. . .<l>„(5)=w, 6 = a«V. . ., a'^a, /S'^/S, . . ., 
the third formula being a generalization of Gauss' (4) . If ^ is a fixed integer 
prime to m, $„(w) is the number of sets of n integers <m and prime to m 
such that each term of a set exceeds by k the preceding term modulo m. 
Consider the productPof the Xth terms of the *J>„(m) sets. If n = 1, P= =t 1 
(mod m) by Wilson's theorem. If n> 1, 
P"-i=)(-l)^-ir-i(X-l)!(n-X)!j*>) (modm). 
For the case A: = X = 1, n = 2, we see that the product of those integers < m 
and prime to m, which if increased by unity give integers prime to m, is 
= 1 (mod m) . 
E. Lucas^^^ gave a generalization of Schemmel's function, without men- 
tion of the latter. Let ei,..., e^ be any integers. Let ^(n) denote the 
number of those integers h, chosen from 0, 1, . . ., n — 1, such that 
h — ei, h — e2,. . ., h — Ck 
are prime to n. For k<n, ei = 0, 62= —I,. .., ei,— — {k — 1), we have k con- 
secutive integers h, h-j-1,. . ., h+k — l each prime to n, and the number of 
such sets is */c(n). Lucas noted that ^(p)'^(g) =^{pq) if p and q are rela- 
tively prime. Let n = a°-h^ . . ., where a, h,. . . are distinct primes. Let X 
be the number of distinct residues oi ei, . . . , e^ modulo a; fx the number of 
their distinct residues modulo h; etc. Then 
^(n)=a»-i(a-X)&^-\6-M). . .. 
L. Goldschmidt^®^ proved the theorems stated by Schemmel, and himself 
stated the further generalization: Select any a — A positive integers <a, 
any h—B positive integers <b, etc.; there are exactly 
a''-\a-A)¥-\h-B)... 
integers <m which are congruent modulo a to one of the a — A numbers 
selected and congruent modulo b to one of the h — B numbers selected, etc. 
P. Bachmann^^^ proved the theorems due to Schemmel and Lucas. 
Jordan's Generalization of Euler's ^-Function. 
C. Jordan,^*^*^ in connection with his study of linear congruence groups, 
proved that the number of different sets of k (equal or distinct) positive 
integers ^n, whose g. c. d. is prime to n, is* 
^^ ■/>.w=»'(i-^.)-..(i-^j 
"ojour. fur Math., 70, 1869, 191-2. "'Th^orie des nombres, 1891, p. 402. 
""Zeitschrift Math. Phys., 39, 1894, 205-212. i^^Niedere Zahlentheorie, 1, 1902, 91-94, 174-5. 
^iioTraitg des substitutions, Paris, 1870, 95-97. 
*He used the symbol [n, k]. Several of the writers mentioned later used the symbol (f>k(n), 
which, however, conflicts with that by Thacker.^^" 
