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144                         HlSTOEY OF THE THEOKY OF NUMBERS.                    [CHAP. V
A. Minine165 investigated the numbers N which divide the sum of all the integers < N and prime to N.
E. Ces&ro166 proposed his theorems160 as exercises. Proofs, by associating a with N a, etc., were given by Moret-Blanc (3, 1884, 483-4).
Ces&ro57 (p. 82) proved the formula of Liouville.154 Writing (pp. 158-9) <f)m for <l>m(N) and expanding <3f>7n=S(JV a)m, where a, 0, . . . are the integers ^N and prime to N, we get
whence <t>m is divisible by N if m is odd, but not if m is even. This is evident (p. 257) since am+(Na)m is divisible by a+N a if m is odd. The above formula gives Am = (1  A)m, symbolically, where
A   =^-1
m    4 Nm
is the arithmetic mean of the rath powers of a/Nt P/N, .... The mean value of <t>m(N) is 6AmNm+l/ir*. He reproduced (pp. 161-2) an earlier formula,160 which shows that J5m=(l )m, symbolically, if 5m is the arithmetic mean of the products of a/JV, /3/^V, - . . taken m at a time. We have (p. 165) the approximation
xm+2         6
whence (p. 261) the mean of </>w(N) is
Proof is givien (pp. 255-6) of Thacker's150 formula
ranging over the divisors of Nt and w over the prime divisors of N.    Here is Merten's function (Ch. XIX).   It is proved (pp. 258-9) that
the first characterizing the function $P(N), and reducing to (4) for p = 0. If a ranges over the integers for which [2n/a] is odd, then (p. 293)
exactly if m = 0, 1, 2, 3, approximately if m> 3, where A is the excess of the sum of the inverses of 1, . . ., n over that of n+1, . . ., 2n.   In particular,
"Math. Soc. Moscow (in Russian), 10, 1882-3, 87-101. 1MNouv. Ann. Math., (3), 2, 1883, 288.