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+(N—a)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 where 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.