Chap. V] GENERALIZATIONS OF EuLER's 0-FuNCTION. 143 
Several ^^^^ found expressions for 0„=<^„(iV) and proved that 
</>ox'»+n</)ia;"-i+i7i(n- 1) (/)2a;"-2+ . . . +<^„= (n odd) 
has the root —4>\/4>q, while the remaining roots can be paired so that the 
sum of the two of any pair is — 20i/(/)o. If n=3 the roots are in arith- 
metical progression. 
H. Postula^^^ proved Crelle's result by the long method of deleting 
multiples, used by Brennecke.^^^ Catalan {ibid., pp. 208-9) gave Crelle's 
short proof. 
Mennesson^^^ stated that, if q is any odd number, 
<^» = *0(^'+') (modg), 
and (Ex.366) that the sum of the products (/)(n) — 1 at a time of the integers 
^n and prime to n is a multiple of n. 
E. Cesaro^^° proved the generalization: The sum rprn of the products m 
at a time of the integers a, ^,. . .^N and prime to N is divisible by iV if m 
is odd. For by replacing a by iV— a, /3 by A^"— j8, . . . and expanding. 
'^^-&^Ht-\y-Mt'-2y 
¥2-... 
where 0=0(iV). Also (l>m{N) is divisible by iV if m is odd. 
F. de Rocquigny^^^ proved Crelle's result. Later, he"^ employed con- 
centric circles of radii 1, 2, 3, . . . and marked the numbers {m — l)N-}-l, 
(m — l)N-\-2, . . ., mN at points dividing the circle of radius m into A'' equal 
parts. The lines joining the center to the 0(iV) points on the unit circle, 
marked by the numbers <N and prime to N, meet the various circles in 
points marked by all the numbers prime to N. He stated that the sum 
of the 4>{N) numbers prime to N appearing on the circle of radius m is 
|(2m — l)0(iV^), and [the equivalent result] that the sum of the numbers 
prime to N from to mN is ^'m^(i>{N^). He later recurred to the subject 
{HUd., 54, 1881, 160). 
A. Minine^®^ noted that, if P>N> 1 and k is the remainder obtained by 
dividing P by N, the sum s{N, P) of the integers <P and prime to N may 
be computed by use of 
s{N, mN+k)=s{N, k)+^4>{N')+mN4>{N)„ 
where (Minine^O <i>{N)k is the number of integers ^k prime to N. 
*A. Minine^^^ considered the number and sum of all the integers < P 
which are prime to N [Legendre's (5) and Minine^*^^]. 
i"«Matli. Quest. Educ. Times, 28, 1878, 45-7, 103-5. 
i"Nouv. Corresp. Math., 4, 1878, 204-7. Likewise, R. A. Harris, Math. Mag., 2, 1904, 272. 
^^lUd., p. 302. 
i6«76id., 5, 1879, 56-59. 
"iLes Mondes, Revue Hebdom. des Sciences, 51, 1880, 335-6. 
i62/6id., 52, 1880, 516-9. 
i"/6id., 53, 1880, 526-9. 
"^Nouveaux theoremes de la th^orie des nombres, Moscow, 1881. 
