PURE MATHEMATICS 347 



(3) The lattice-points of a right-angled triangle. 



If CO, 0)^ are two numbers whose ratio 6= co/co'^ is irrational, 

 and J denote the triangle whose sides are the co-ordinate 

 axes and the line 



cox + co^y = 'J7>0 



and N('7) the number of lattice-points {Gitterpunkte , i.e. points 

 whose co-ordinates are integral) which lie inside A, then how 

 accurate an approximation can we find for N(77) when 77 is 

 large ? And how does the accuracy of the approximation 

 depend upon the arithmetical character of ^ ? This problem 

 (Problem A) has been treated by Hardy and Littlewood in 

 two papers {Proc. L.M.S., 20, 192 1, 15-36 ; Abhl. math. Sent. 

 Hamburg, 1, 1922, 212-49) ; it is related to another problem 

 (Problem B) which had been considered before by Lerch, 

 Weyl, and the authors themselves. If {x] denote the integral 

 part of X, and {a^} = .t— [.r] — \, then what is the most that 

 can be said as to the order of magnitude of the sum 



s{e,n) = X{ve\ 



v = l 



when n is large ? It is shown, in fact, in the first paper men- 

 tioned that 



where 8(77) is a sum very similar to s{6, n). 



In the first paper, Hardy and Littlewood prove that 

 8(77) = 0(77) for any irrational 6 ; and that this is the most 

 that is true for every irrational. If, however, the quotients 

 in the continued fraction for 6 are bounded, then 8(77) = 0(log77), 

 and this too is a best possible result of its kind. These results 

 are proved by elementary arguments ; the remainder of the 

 paper is based on the properties of a particular analytic func- 

 tion, a degenerate form of Barnes' Double Zeta-function. By 

 its means it is shown that if 6 is algebraic then 8(77) = 0(77"), 

 where a< i . 



Various of these results were proved independently a little 

 later by E. Hecke {Abhl. math. Sem. Hamburg, 1, 1922, 54-76), 

 who used transcendental methods, different from the above, 

 throughout, and by A. Ostrowski {ibid., 77-98, 250-1), with 

 elementary methods. 



Hardy and Littlewood's second paper contains proofs of 

 various further results, some of which had been enunciated 

 in an appendix to the first paper. In particular, they show 

 that if there are constants /f>i and A>0 such that 

 w^lsin w^7r|>A for all positive integral values of n, then 



