Chap. V] GENERALIZATIONS OF EuLER's </)-FuNCTION. 145 
P. Nazimov^" (Nasimof) noted that, when x ranges over the integers 
^m and prime to n, the sum of the values taken by any function /(x) equals 
\mld\ 
7:ix{d)Xf{dx), 
d 1=1 
where d ranges over all divisors of n. The case f{x) = 1 yields Legendre's 
formula (5) . The case/(a;) = xyields a result equivalent to that of Minine.^^^"'* 
A generalization was given by Zsigmondy'^'^ and Gegenbauer.^'^^ 
E. Cesaro^^^ noted that, if A^ is the arithmetic mean of the mth powers 
of the integers ^ N and prime to N, and B^ that of their products m at a 
time, we have the symbolic relations 
Cesaro^^^ proved Thacker's^^'^ formula expressed as 
the last being symboHc, where f^ is a function such that l^^,,{d)='n}~^, d 
ranging over the divisors of n. By inversion 
n(n)=2M©<i'-'=;^n (!-»-'), 
where u ranges over the distinct prime factors of n. 
L. Gegenbauer^'^° proved that, if j/= yln , 
x-l n=lL3; J ,-1 
For the case /c = 0, p = 2, this becomes Bougaief 's^^^ formula 
ig2{x)=i\^^<i>{x), v = [Vn]. 
C. Leudesdorf^'^^ considered for fx odd the sum i/'^(iV) of the inverses of 
the juth powers of the integers < N and prime to N. Then 
^|^,{N)=^kN'-hfiN^P,+,iN), 
where k is an integer. Thus, if N = p^q, where q is not divisible by the 
prime p>3, »/'^(A^) is divisible by p^' unless ju is prime to p, and )U+1 is 
divisible by p — 1; for example, \{/^{p) is divisible by p^. If p = 3, ^l/^iN) 
is divisible by p^' if fx is an odd multiple of 3. If p = 2, it is divisible by 
2^'~^ except when q = l. 
Cesaro"^ inverted his" symbolic form of Thacker's formula for <l)m{N) 
in terms of xf/'s and obtained 
nB,rPp{n) = {<f>-nBy. 
i"Matem. Sbomik (Math. Soc. Moscow), 11, 1883-4, 603-10 (Russian). 
"'Mathesis, 5, 1885, 81. 
"'Giomale di Mat., 23, 1885, 172-4. 
""Sitzungsber. Ak. Wiss. Wien (Math.), 95, II, 1887, 219-224. 
i"Proc. London Math. Soc, 20, 1889, 199-212. 
"Teriodico di Mat., 7, 1892, 3-6. See p. 144 of this history. 
