338 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. XI 



the sum o- M (&) of the ^th powers of the divisors of k and given in Ch. X of 

 Vol. I of this History. From the above formula we readily pass to 



V. A. Lebesgue 17 noted that for any integer m, 



m) - (6m - 



which reduces for the case m a prime to the final formula (H') of Liouville. 2 

 Liouville 18 gave formulas of the type of those in his series of articles. 

 Liouville 19 noted that, for any integer m, 



[Vm] 



w) + 2 2 ( m 5mJ)?i(tfi TI) = or w(4m 



according as m is not or is a square. This follows from (<) of Liouville 8 

 with F(z, ?/, 2) = xyz. 



H. J. S. Smith 19 " gave a proof of (a) and 



the summations extended respectively over all solutions of 



m = 2m' 2 + d'8', 2m = TT?" + did 1} 



where d f , S', di, d l} m, mi are positive and odd, while f(x) is an odd 

 function. 



C. M. Piuma 20 proved (e), (L), (N), (7), and 0). 



E. Fergola 21 stated and G. Torelli 22 proved a theorem related to one in 

 Liouville's seventh article. Let a n denote the product of the highest 

 power of 2 dividing n by the sum of the odd divisors of n. Then 



a n 2a n -i + 2a n _ 4 2a n _ 9 + 2a n _ 16 2a n _ 25 + = ( I)"" 1 n or 0, 



according as n is or is not a square. 



S. J. Baskakov 23 proved the formulas in Liouville's twelfth article. 



T. Pepin 24 proved all the formulas in Liouville's first five articles except 

 (f) and its specializations (H), (g). 



N. V. Bougaief 25 proved some of the theorems in Liouville's series of 

 articles by showing that, if F(x) is an even function, an identity 



oo oo 



X) A m cos mx = X) Bn cos nx 



m=0 n=0 



implies SA m F(m) = SB n F(ri), and a similar theorem involving sines and 

 an odd function Fi(ri). 



17 Jour, de Math., (2), 7, 1862, 256. 



18 Ibid., 41-8. To be considered under class number in Vol. III. 



19 Jour, de Math., (2), 7, 1862, 375-6. 



190 Report British Assoc. for 1865, art. 136; Coll. Math. Papers, I, 346. 

 20 Giornale di Mat., 4, 1866, 1-14, 65-75, 193-201. 



21 Giornale di Mat., 10, 1872, 54. 



22 Ibid., 16, 1878, 166-7. 



23 Math. Soc. Moscow, 10, I, 1882-3, 313. 



24 Atti Accad. Pont. Nuovi Lincei, 38, 1884-5, 146-162. 

 26 Math. Soc. Moscow, 12, 1885, 1-21. 



