CHAPTER XI. 



LIOUVILLE'S SERIES OF EIGHTEEN ARTICLES. 



J. Liouville enunciated without proof numerous results in a series of 

 eighteen articles, " Sur quelques formules ge'ne'rales qui peuvent etre utiles 

 dans la the"orie des nombres." 



Let m be an odd integer, a an integer i= 1. Set 



2 a m = m' + m", m = dd, m' = d'd', m" = d"8", 



where m' and m" are odd positive integers. Let/(x) = /( x) be an even 

 single- valued function. He 1 stated that 



(a) Z ( Z [/(<*' ~ O - f(d' + d")]} = V- 1 Z d\m - /(2-d) }, 



d', d" d 



where d, d', d" range over all the divisors of m, m', m", respectively, and 

 the first summation extends over all the pairs of positive odd integers m r t 

 m" whose sum is 2 a m. Taking f(x) = x*, we get 



where the coefficients are those of even rank in the binomial formula, and 

 f M (m) denotes the sum of the nth powers of the divisors of m. For n = 1 

 and IJL = 2, we have 



2 3a - 3 r 3 (m) = ZMwOMm"), 2 5a - 5 r 5 (m) = 2 Mm') Mm"). 



The first gives the number of decompositions of 4-2 a m as a sum of 8 odd 

 squares; the second gives the number of decompositions of 8-2m into 

 s + 2<r, where s is a sum of 8 odd squares such that s/8 is odd, while a is a 

 sum of 4 odd squares. 



For f(x) = cos xt, (a) gives 



2(2 sin d't 2 sin d"f) = 2 a ~ 1 2rf sin 2 (2- 1 dZ). 

 Taking a = 1, t = x/2, or by setting /(x) = ( l) z/2 , we get 

 2(2(- l) (d '- 1)/2 - 2(- l)^'- 1 ^ 2 ) = 2d = TiW, 



which yields Jacobi's 25 - 30 theorem of Ch. VIII that 4w has Ti(w) representa- 

 tions as a sum of four odd squares. 



For a function f(x, y) which is unaltered by the change of the sign of x 

 or of y, Liouville stated that 



Z I _..[/(<*' - d", 8' + 8") - f(8' + 6", d' - d 



(b) \^ 



(c) Z ( Z U(d' ~ d", d' + 5") - f(d' + d", d' - 5")]} = * 



d', d" 



1 Jour, de Math., (2), 3, 1858, 143-152, 193-200. First and second articles. 



329 



