144 History of the Theory of Numbers. [Chap, v 
A. Minine^^ investigated the numbers N which divide the sum of all 
the integers < N and prime to N. 
E. Cesaro^^^ proposed his theorems^®^ as exercises. Proofs, by associa- 
ting a with N — a, etc., were given by Moret-Blanc (3, 1884, 483-4). 
Ces^ro" (p. 82) proved the formula of Liouville.^" Writing (pp. 158-9) 
<f)„ for </),„(A0 and expanding 0„=2(iV— a)"*, where a, /3, . . . are the integers 
^ A^ and prime to A'^, we get 
whence <^^ is di\'isible by N if m is odd, but not if m is even. This is e\'ident 
(p. 257) since aJ^ -{- {N — a)"" is di\isible by a+A'' — a if m is odd. The above 
formula gives A'" = (1 — A)"*, symboUcally, where 
" 4> AT-" 
is the arithmetic mean of the mth powers of a/N, ^/N, .... The mean 
value of <j)m{N) is 6A„A"'"+V'''"^- He reproduced (pp. 161-2) an earHer for- 
mula,^^° which shows that B"' = {l-B)'", symbolically, if B^ is the arith- 
metic mean of the products of a/N, ^/N, . . . taken m at a time. We have 
(p. 165) the approximation 
X x"*"^^ 6 
2 <f)m(j) = 7 — rrr} — r^ * ~2> 
y=i (7M+l)(m-|-2) tT 
whence (p. 261) the mean of (t>^{N) is 6A''"+7(m+l)7r2. 
Proof is given (pp. 255-6) of Thacker's^^° formula 
*-«'" '"'*'::;"'*'"" -iiiij.cr)'-»-""'"' 
where 
UN)=^d'-'f^(d)=Il{l-u^-'), 
d ranging over the divisors of A^, and u over the prime divisors of N. Here 
nix) is Merten's function (Ch. XIX). It is proved (pp. 258-9) that 
2d^-Vp(^ = 1, 2^V.(^ =2dV,-.(rf), 
the first characterizing the function \pp{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 7?7 = 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 + l, . . . , 2n. In particular, 
20(a) =nl 
'"Math. Soc. Moscow (in Russian), 10, 1882-3, 87-101. 
»«Nouv. Ann. Math., (3), 2, 1883, 288. 
