302 Mr Viggo Brun, On the function \x'] 



We can express 77 {s) in terms of log t, {s) by using the factors fx {n) 

 of Mobius, but we soon discover that it is more practical to study 

 the function 



f{x) = 7r {x) + Itt [Vx) + In (^y^) + I77 {.^/x) + ... 



than the function n {x) itself; and it is not difficult to deduce the 

 formula 



77 (a;) + ^{\/^+ ... 



_ log g js) x^ log i (2 s) x2« log I {2>s) x^^ . 



- J 2! + 3! ••• + ^ •••(3). 



The function / [x) has points of discontinuity only for integral x\ 

 and R" can be given the same form as R, viz. 



R" ^ 8^2-«/^+^ 

 It follows that 



77 [x) + I77 (V^) + ... = lim S (- l)«+i ^Qg ^ (^^) ^^' __(4^ 

 s -*■ 00 ?i = 1 w ! 



for all non-integral x. 



This formula has been deduced by Helge von Koch, in Vol. xxiv 

 of the Acta Mathefnatica, but is not mentioned in the Handhuch 

 of Landau nor in the Encydopedie des sciences mathematiques 

 (Pt I, Vol. Ill, Fasc. 4). 



It is very interesting to compare the formulae (1) and (3). The 

 only difference between them is that (1) has ^ (s) where (3) has 

 log C (s), and in spite of this the formula (1) gives an approximation 

 for the regular function [x] while the formula (3) gives an approxi- 

 mation for the very irregular function / (a;) of Eiemann. 



§ 5. If we use for our function O (x) the formula of Kronecker 

 (see Encycl. des sciences math., Pt I, Vol. iii, Fasc. 3, p. 256) 



"I fa-\-iao ^s 



we obtain the famous formula of Riemann 



which is equivalent to that of Helge von Koch (4). We equally 

 obtain the following formula, equivalent to (2), 



1 r«+^=° X' C (s 

 valid for all positive non-integral x 



f^]-2«L....^^'''» (5)^ 



