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CHAP. V] GENEEALIZATIONS OF EULEB'S ^-FUNCTION. • 147
SCHEMMEL'S GENERALIZATION OF EULER'S ^-FUNCTION.
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
h—e1} h—e2,. . ., h—ek
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 b—B numbers selected, etc. P. Bachmann193 proved the theorems due to Schemmel and Lucas.
JORDAN'S GENERALIZATION OF EULER'S ^-FUNCTION.
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*
lfl°Jour. 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°