CHAP. V] GENERALIZATIONS OF EULEH'S ^-FUNCTION. 143 Several1670 found expressions for <!>n~(t>n(N) and proved that <£ozn+^i^~1+itt(tt-l) 02xn-2+ . . . +0n= 0 (n odd) has the root — <£i/<£o, while the remaining roots can be paired so that the sum of the two of any pair is — 2<£i/<£0. If n= 3 the roots are in arithmetical progression. H. Postula158 proved Crelle's result by the long method of deleting multiples, used by Brennecke.152 Catalan (ibid., pp. 208-9) gave Crelle's short proof. Mennesson159 stated that, if q is any odd number, and (Ex. 366) that the sum of the products <j>(n) — 1 at a time of the integers ^ n and prime to n is a multiple of n. E. Cesaro160 proved the generalization: The sum i/>m of the products m at a time of the integers a, /3, . . . ^ N and prime to N is divisible by AT if m is odd. For by replacing a by N— a, ft by N— 0, . . . and expanding, where </>==</> (j/V). Also 4>m(AT) is divisible by N if m is odd. F. de Rocquigny161 proved Crelle's result. Later, he162 employed concentric circles of radii 1, 2, 3, . . . and marked the numbers (m — 1)2V+1, (m — 1) AT 4-2, . . . , mN at points dividing the circle of radius m into N equal parts. The lines joining the center to the 4>(N) 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 <t>(N) numbers prime to N appearing on the circle of radius m is |(2m — !)</>( N2), and [the equivalent result] that the sum of the numbers prime to N from 0 to mN is %m2(j>(N2). He later recurred to the subject (*ibid., 54, 1881, 160). A. Mim'ne163 noted that, if P>N> I 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 2 »(N, mN+k)=s(N, fc)+^(ff«) +«#*(#)», « ^ where (Minine47) <l>(N)k is the number of integers ^k prime to N. *A. Minine164 considered the number and sum of all the integers <P which are prime to N [Legendre's (5) and Minine163]. »««Math. Quest. Educ. Times, 28, 1878, 45-7, 103-5. 158Nouv. Corresp. Math., 4, 1878, 204-7. Likewise, R. A. Harris, Math. Mag., 2, 1904, 272. «»/6iU, p. 302. ™Ibid., 5, 1879, 56-59. 161Les Mondea, Revue Hebdorn. des Sciences, 51, 1880, 335-6. uzlbid., 52, 1880, 516-9. ™Ibid., 53, 1880, 526-9. 164Nouveaux thdoremes de la the"orie des nombres, Moscow, 1881.