CHAP. IX] SUM OF FlVE OR MORE SQUARES. 317 



B. Boulyguine 52 employed the notations 



(4AA 

 2 J 



& 

 Z (TO) = 



summed for all the N p (ri) integral solutions (positive, negative, or zero) 

 of x\ + + x 2 p = n. Write ff k (m) for the sum of the kth powers of the 

 divisors of m and 



Pk (m) = S(- 1 



for the difference between the sum of the kth powers of the divisors 4/i + 1 

 of m and the sum of the kth powers of the divisors 4h + 3. By use of 

 elliptic functions, it is shown that, if n = 2 a m, where m is odd, 



(n) + a ()+" + a< r " Z 



8r-6 8r 14 



There is given a similar expression for N &r +6(n). Also, 



24r+3(l+a) _ 24H-4 I J 



24r+ 



3 



Z (n) + < 2) Z (n) + - - - + < r) Z (n), 



Sr 8r-8 8 



with a similar expression for A^ 8 r+4(w). Here the a's and d's are rational 

 numbers not depending on n. It is stated that there result the known 

 formulas for the number of decompositions into 2, 4, 6, 8, 10, or 12 squares 

 and apparently new formulas for 14 or 16 squares. 



Boulyguine 53 stated a recursion formula for his 52 S(w): 



ArN r (n) = F r (n) + A rl Z (n) + ^ Z (n) + A rZ Z (w) + -, 



r-16 r-24 



for r = 2, 3, , where A r , A r i, - are independent of n, while F r (ri) is a 

 specified function differing in the three cases r odd, r = 4k + 2, r = 4k + 4. 

 S. Ramanujan 54 studied the function ^(n) for which 



Z 



n=0 



Special cases of ^ are the functions x(w), P(w), X4(n), 0(w), Q(n) of Glaisher" 

 of Ch. VIII. He touched (pp. 179, 183-4) on the number of representations 

 of n as a sum of s squares, s = 10, 16, etc. 



L. J. Mordell 55 proved that various empirical results of Ramanujan 54 

 follow from expansions of elliptic modular functions. 



62 Comptes Rendus Paris, 158, 1914, 328-330. 

 53 Ibid., 161, 1915, 28-30. 

 M Trans. Cambr. Phil. Soc., 22, 1916, 173-9. 

 65 Proc. Cambr. Phil. Soc., 19, 1917, 117-124. 



