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CHAP. V]             GENEEALIZATIONS OF EULEB'S ^-FUNCTION.                 147
V. Schemmel190 considered the $n(w) sets of n consecutive numbers each <m and relatively prime to m. Ifm = aa^. . . , where a, 6, . . . are distinct primes, and m, m' are relatively prime, he stated that
the third formula being a generalization of Gauss' (4) . If k is a fixed integer prime to m, $n(w) is the number of sets of n integers <w and prime to m such that each term of a set exceeds by k the preceding term modulo w. Consider the productPof the Ath terms of the $n(m) sets. If n = 1, PS =t l (mod m) by Wilson's theorem. If n> 1,
For the case fc=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 s= 1 (mod m) .
E. Lucas191 gave a generalization of SchemmeFs function, without mention of the latter. Let 61, . . . , ek be any integers. Let St'(n) denote the number of those integers h, chosen from 0, 1, . . ., n 1, such that
he1} he2,. . ., hek
are prime to n. For k<n, e^O, e2=  !>  > fy,=  (fc  1), we have k consecutive integers ft, h+lt. . ., /i-ffc  1 each prime to n, and the number of such sets is ^(n). Lucas noted that ^f(p)^f(q) =^(pq) if p and g are relatively prime. Let n = aab|9. . ., where a, 6, . . . are distinct primes. Let X be the number of distinct residues of e1} . . ., ek modulo a; ju the number of their distinct residues modulo 6; etc. Then
L. Goldschmidt192 proved the theorems stated by Schemmel, and himself stated the further generalization: Select any a A positive integers <a, any 6 B positive integers <b, etc.; there are exactly
integers <m which are congruent modulo a to one of the a A numbers selected and congruent modulo 6 to one of the bB numbers selected, etc. P. Bachmann193 proved the theorems due to Schemmel and Lucas.
C. Jordan,200 in connection with his study of linear congruence groups, proved that the number of different sets of k (equal or distinct) positive integers gn, whose g. c. d. is prime to n, is*
lflJour. fiir Math., 70, 1869, 191-2.                    1MThdorie des nombres, 1891, p. 402.
l92Zeitschrift Math. Phys., 39, 1894, 205-212.      193Niedere Zahlentheorie, 1, 1902, 91-94, 174-5. *Trait6 des substitutions, Paris, 1870, 95-97. *He used the symbol [n, k].   Several of the writers mentioned later used the symbol <fc(n),
twViinK   hrkwovpr   nymfliot.Q with that, hv Tlianlrftr 16