Chap. X] SuM AND NuMBER OF DiVISORS. 285 
where d ranges over the divisors of m. He proved (p. 411) that 
S(-l)'"/'^fi = 2(T(m/2)-(7(m). 
J. Liouville^^ stated without proof the following formulas, in which d 
ranges over all the divisors of m, while 5 = m/d : 
Xaid) =2:5r(d), S0(d)r(5) =(r(m), 20(d)r(6) = [rim)}^, 
XcT{d)cr{8) =SdT(d)r(5), Sr(d)r(5) =2:|t(^)}' 
where (}){d) is the number of integers < d and prime to d, 6{d) is the number of 
decompositions of d into two relatively prime factors, and the accent on S 
denotes that the summation extends only over the square divisors D^ of m. 
He gave (p. 184) 
S0(cf)=r(m2), ^'e{^)i=r{m), 
the latter being implied in a result due to Dirichlet.^^ 
Liouville"*' gave the formulas, numbered (a),. . ., {k) by him, in which 
X(m) = +1 or —1, according as the total number of equal or distinct prime 
factors of m is even or odd: 
Sr(d2'')=T(m)r(m''), 2r(d2'')T(5) =ST(d)rOT, S(^(5)(7((i) =mT(m), 
S5(7(d) =SdT(d), SX(c^) = 1 or 0, ^\{d)d{d)r{b) = 1 or 0, 
according as m is or is not a square; 
i:\{d)d{d)r{h^) = l, X\{d)e{d)=\im), SX(d)0(5) = l, 
l^X{d)d{d)did)=0, SX(5)o-(ci) =mS'-^. 
The number of square divisors D^ of m is '2\(d)T{8). 
Liouville^^ gave the formulas, numbered I-XVIII by him: 
ST(52)(/)(d) =S5^(d), Sdr(52) =S^(5)(r(d), 
ST(52)X(d) =T(m), 2 {T{8)}Md)d{d) =tW, 
S0(d)T(5)r(5'') =SdT(62''), ^e{b)T{d)r{d'') =Sr(52)T(d-''), 
ST(52'')(7(d) =25r(d)T(d''), S'0(i))T(^) =S'Z) ^(^) , 
SX(5)T(d)TW =2't(^) . ^\id)(T{d) =mX(m)S'- 
2)2 
'^Jour. de Mathematiques, (2), 2, 1857, 141-4. "Sur quelques fonctiona num^riques," 1st article. 
Here Sa6c denotes S(a6c). 
^^Ihid., 244-8, second article of his series. 
"76id., 377-384, third article of his seriea 
