98 HISTORY OF THE THEORY OP NUMBERS. [CHAP. II 



Hardy and Littlewood 276 also considered the same problem when n l 

 is replaced by an arbitrary increasing sequence \ n with infinite limit and 

 obtained the same result, but with the exception of a set of values of 6 of 

 measure zero. R. H. Fowler 2776 established uniform distribution, with an 

 upper bound for the error, when X n increases as rapidly as an exponential 

 e nS (8 > 0). Weyl 277a extended the theorem of uniform distribution to all 

 cases in which X n increases with tolerable regularity and as fast as (log n) 2H 

 (5 > 0). These questions are intimately connected with the problem of 

 the behavior of the series Sf exp. (27rt'X n ) when N -> < , which has been 

 considered in detail by Hardy and Littlewood, 2770 and Weyl. 277a 



G. Giraud 278 proved that there exist integral values of the z's and i/'s 

 for which 



I Xi any i a ip y p Ai\ < e (i = 1, , n), 



whatever e be, if and only if all the forms miXi + + m n X n , which take 

 integral values when Xi, , X n are replaced in turn by the p sets of 

 values aijj - -, a nj - (j = 1, , p), take also integral values when we replace 

 Xi, --,X n by Ai, , A n . 



S. L. van Oss 279 proved that n real linear functions of x it , x n of deter- 

 minant unity have the minimum value unity for integral z's if at least one 

 of the forms has integral coefficients without a common divisor. This 

 had been proved by Minkowski for n = 3. 



W. E. H. Berwick 280 gave a method to find which pair of integers 

 x, y (0 ^ y < N) gives the least value for / = ax + by + c, where a, 6, c 

 are real and not zero. Thus he finds the point with integral coordinates 

 nearest to the line / = and within the strip between y = and y = N. 



A. Brown 281 noted that, to find the fraction whose denominator does 

 not exceed a given integer and which approximates most closely to a given 

 number, Lagrange's theory gives the fraction nearest in defect and the 

 fraction nearest in excess, but does not decide which of them is nearest in 

 absolute value to the given number. A simple method is here given for 

 deciding between the two fractions. 



A. J. Kempner 282 noted that any straight line with an irrational slope 

 has on either side of it an infinitude of points with integral coordinates 

 lying closer to it than any assigned distance. 



G. Humbert 283 developed Hermite's 259 method of approximating to an 

 irrational number co, showed that it differs very little from the method of 



278 Acta Math., 37, 1914, 155-191; Proc. Fifth Internat. Congress Math., 1, 1912, 223-9. 



277 Proc. London Math. Soc., (2), 16, 1917, 294-300. 



2770 Gottingen Nachrichten, 1914, 234-244; Math. Annalen, 77, 1916, 313-352. 



2 Proc. London Math. Soc., (2), 14, 1915, 189-206. 



277c Acta Math., 37, 1914, 155-238; Proc. Nat. Acad. Sc., 2, 1916, 583-6; 3, 1917, 84-8. 



278 Soc. Math. France, Comptes Rendus des Stances, 1914, 29-32. 



279 Handelingen XVde Nederlandsch Natuur- en Geneeskundig Congres, 1915, 192-3. 



280 Messenger Math., 45, 1916, 154-160. 



281 Trans. Phil. Soc. South Africa, 5, 1916, 653-7. 



282 Annals of Math., 19, 1917, 127. 



283 Comptes Rendus Paris, 161, 1915, 717-21; 162, 1916, 67; Jour, de Math., (7), 2, 1916, 



79-103. 



