Chap. V] EulEr's 0-FuNCTION. 131 
For d = n,r' = l,r = v — l, this becomes the former result ; f or r = r ' = 1 , 5 = n, 
it becomes 20 (x) =n^, where x takes the values for which p(n/rc)^ 1/2. 
H. W. Lloyd Tanner^^ studied the group G of the totitives of n (the 
integers <n and prime to n), finding all its subgroups and the simple groups 
whose direct product is G. 
E. Lucas^^ proved that, in an arithmetical progression of n terms whose 
common difference is prime to n, there are (ji{d) terms having with n the g. c. d. 
n/d. If, when d ranges over the divisors of n, Xxpid) =n for every integer n, 
then (p. 401) \p{n)=(^{n), as proved by using n = l, a, a^,. . ., and n = ah, 
a^b, . . . , where a,h,. . . are distinct primes. He gave (pp. 500-1) a proof of 
Perott's^^ first formula by induction from N — 1 to N, communicated to him 
by J. Hammond. The name " indicateur "of n is given (preface, xv) to 
<f){n) [Prouhet^sj. 
C. Moreau (cf . Lucas,'^^ 501-3) considered the C{n) circular permutations of 
n objects of which a are alike, (3 alike, . . . , X alike. Thus, if a = 2, /3 = 4, the 
C(6) = 3 distinct circular permutations are aahbhb, ababbb, abbabb. In general, 
^^^^=n^^^^^(a/d)!...(X/d)r 
where d ranges over the divisors of the g. c. d. of a, jS, . . . , X. In the 
example, d = 1 or 2, and the terms of the sum are 15 and 3. 
P. A. MacMahon^^ noted that C(n) = 1 if n = a, so that we have formula 
(4). His expression for the number of circular permutations of p things n 
at a time is quoted in Chapter III on Fermat's theorem. 
A. Berger^^" evaluated S^il k'^%{k). For a = 2 the result is 3nV7rH 
\n log n, where X is finite for all values of n. 
E. Jablonski'^^ considered rectilinear permutations of indices a, . . ., X, 
with the g. c. d. D. Set a = a'D,- • .,\ = \'D, a+ • . .+X = m = m'Z). Then 
the number of complete rectilinear permutations of indices a'n, . . . , \'n is 
P{n)=- ^'^'''^' 
{a'n)\...{\'n)\ 
The number of complete circular permutations is 
where d ranges over the divisors of D. If Q{D/d) is the number of rectilinear 
permutations of indices a, . . . , X which can be decomposed into d identical 
portions, ^Q(D/d)=P{D). Also 
'iProc. London Math. Soc, 20, 1888-9, 63-83. 
"Theorie des nombres, 1891, 396-7. The first theorem was proved also by U. Concina, II 
Boll, di Matematica, 1913, 9. 
"Proc. London Math. Soc, 23, 1891-2, 305-313. 
'»«Nova Acta Regiae Soc. Sc. UpsaUensis, (3), 14, 1891, No. 2, 113. 
'♦Comptes Rendus Paris, 114, 1892, 904-7; Jour, de Math., (4), 8, 1892, 331-349. He proved 
Moreau's'* formula for C{n). 
