300 Mr Viggo Brun, On the function [x] 



is erroneous, the correct asymptotic formula being according to 

 Mertens 



'n{x)-7r {^~x) +l = k.{l-l){l-l)...{l-~y, 



where Z^ = ig^^ . 89..., 



C being the Eulerian constant. For this, and also for other reasons, 

 I have multiplied the formula in question by an undetermined 

 constant. I also attempted to approximate to this constant em- 

 pirically; but the value 1'598 thus obtained was seriously in error. 

 The right value r320 has been determined independently byi 

 Stackel {Sitz. der Heidelberger AJcad., Abth. A, 1916) and by Hardy 

 and Littlewood (I.e. supra). 



§ 2. We have seen an example of the error of putting [x] = x. 

 Let us study the function [x] more closely. If we draw the curve 

 y = [x], we see that it forms a discontinuous hne like the steps of 

 a staircase. The same can be said of the curve y = tt {x), but here 

 the steps are not regular. 



Let us try to express our discontinuous function by another 

 discontinuous, but simpler, function. We write 



^ (") = jo (0 <x < 1). 



We see that 



M==<d(x) + o(|) + o(|) + (d(|) + ...; 



e.g. [3 . 5] = O (3 . 5) + (D (^) + O (^^^ == 1 + 1 + 1 =^ 3. 



This formula has been employed by Lipschitz in the Comptes 

 Rendus of 1879. But our function will also express other dis- 

 continuous functions, such as tt (x) : thus 



e.g. 77 (6) = O (I) + O (|) + O (^) = 1 + 1 + 1 = 3. 



§ 3. Let us try to approximate to the function O by a con- 

 tinuous function. We will choose the function 



~ 1 2! ^ 3! ■"' 



where s is a positive integer not less than 2. It is not difficult to 

 see that 1 — e"*' is very near when x < 1 and very near 1 when 



