Mr Viggo Brun, On the function [x] 301 



a:; > 1. At the point of discontinuity x= I, 1 — e~^' = 1 — e-^. We 

 are now able to approximate to the function [x] : thus 



+ ... + R, 



or L^J- J 2] + 3l •. + K. 



It is not difficult to find an expression for R, valid for all values 

 of X which are not nearer to the integers (the points of discon- 

 tinuity of [x]) than e, where e < |. We can give R the following 

 form 



R=^ 8d2-''f^ + \ 



where — 1 < ^ < 1 and where s > x, s > 2. 



We can now give our formula the following two forms 



, , C(s)x' I (2s) x2^ I (3s) a;3« „^^ , ^, 



M^^^^Y ^^21 — + ^] ... + 8^2-^/^+1 ...(1), 



valid for all positive values of x not in the intervals (1 — e, 1 + e), 

 (2 - e, 2 + e), (3 - e, 3 + e), etc.; or 



^-i .LA 1 2! + ^^n~ •••; 



= lim i (- l)n-i ^ -(^-^ ...(2), 



valid for all non-integral x. 



If we wish to deduce results from the method of Eratosthenes, 

 it is advantageous to make use of these formulae. Later on I shall 

 perhaps show some of these applications. 



§ 4. We are also able to find an approximate formula for tt (x), 

 viz. 



y^ (s) x^ 7jJ2s)x^ r j (3s) x^^ 

 77 {x) = -^ 21 + 3l ••• + ^ , 



/ V 1 1 1 1,1, 

 where r) (s) = 2"^ "^ 3^^ "^ 5"^ ^ 7 ^ ^ IP *■" 



This sum can be expressed in terms of ^ (s) by employing the 

 identity of Euler, 



from which results 



log n«) = ^ («) + h (25) + h (3s) + Irj (4s) + .... 



VOL. XX. PART III. 20 



