CHAP. V] GENERALIZATIONS OP EULER'S ^-FUNCTION. 145 P. Nazimov167 (Nasimof) noted that, when x ranges over the integers ^ m and prime to n} the sum of the values taken by any f unction /(x) equals d where d ranges over all divisors of n. The case f(x) = 1 yields Legendre's formula (5). The case/(x) ^xyields a result equivalent to that of Minine.163""4 A generalization was given by Zsigmondy77 and Gegenbauer.173 E. Cesaro168 noted that, if Am is the arithmetic mean of the mth powers of the integers ^N and prime to N, and Bm that of their products m at a time, we have the symbolic relations Am=(N-A)m, Bm = (N-B)m. Cesaro169 proved Thacker's150 formula expressed as the last being symbolic, where f* is a function such that ranging over the divisors of n. By inversion where u ranges over the distinct prime factors of n. L. Gegenbauer170 proved that, if v = ^In , n it r**j ~"| ""5* j 1 fe i ofe i i f f, /xvA \fc I 'V I I JL /M\ ft (t*\ *i *\ Zi jl -r^ -r . . . "T(^PWJ f = ^ -7 19*W> ^plPi • • -P, *-i n-iLa:J For the case /: = 0, p = 2, this becomes Bougaiefs165 formula C. Leudesdorf171 considered for ju odd the sum $»(N) of the inverses of the /zth powers of the integers <N and prime to N. Then where k is an integer. Thus, if N = plq, where q is not divisible by the prime p>3, ^^(N) is divisible by p21 unless /z is prime to p, and JLC+! is divisible by p — 1; for example, ^M(p) is divisible by p2. If p = 3, ^M(-/V) is divisible by p21 if ^ is an odd multiple of 3. If p = 2, it is divisible by 221™1 except when g= 1. Cesaro172 inverted his67 symbolic form of Thacker's formula for 4>m(N) in terms of \^'s and obtained »7Matem. Sbornik (Math. Soc. Moscow), 11, 1883-4, 603-10 (Russian). "•Mathesis, 5, 1885, 81. ""Giornale di Mat., 23, 1885, 172-4. 17°Sitzungsber. Ak. Wiss. Wien (Math.), 95, II, 1887, 219-224. 171Proc. London Math. Soc., 20, 1889, 199-212. 17aPeriodico di Mat., 7, 1892, 3-6. See p. 144 of this history.