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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
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.