Mr Viggo Brun, On the function [x'] 299 



On the function [x\. By Viggo Brun (Drobak, Norway). 

 (Communicated by Prof. G. H. Hardy.) 



[Read 24 January 1921.] 



§ 1. In the Proceedings of the Cambridge Philosophical Society 

 (Vol. XIX, Part 5, 1919) Mr Shah and Mr Wilson have discussed 

 some formulae proposed for calculating the number of Goldbachian 

 decompositions of an even number. They have also mentioned a 

 formula of mine, saying: "The formula to which Brun's argument 

 leads is ...(11)" (page 24-3). In reality I have not enunciated this 

 formula, as Mr Hardy and Mr Littlewood justly remark in their 

 "Note on Messrs Shah and Wilson's paper." 



But it may appear at first sight as if my method should naturally 

 lead to the formula (11), and I should Hke to explain why one 

 will not find it so on examining the matter further. 



The formula in question is deduced from the sieve method of 

 Eratosthenes, employed twice. We will here simplify the question, 

 examining only the common sieve of Eratosthenes. Let us deter- 

 mine the number of primes under 14. We write the 14 numbers 



1 2^ ^ £ '^ ± 7 8 9^ 10 11 12 13 14 



effacing first the numbers 2, 4, 6, 8, 10, 12, 14 and then the numbers 

 3, 6, 9, 12. The uneffaced numbers are the number 1 and the 

 primes between \/l4 and 14. We find that 



IT (14) - 77 (Vl4) + 1 - 14 - 



+ 



"14 



2^j 



"141 _ ri4 



2 J [J 



= 14 -7-4 + 2 = 5, 

 where 77 [x) denotes the number of primes not exceeding x, and 

 \x] denotes the number of integers not exceeding x. This formula 

 can easily be generalised; it is not diSicult to see how*. We could 

 now say that it was "natural" to put [x] = x, and we should then 

 get the approximate formula 



77 (14) - 77 (Vl4) + 1 = 14 - y - y + II = (1 - 1) (1 - 1) 14; 



and generalising, we should get 



77 (X) - 77 (Va") + 1 = (1 - 1) (1 _ 1) ... [1 - A] X, 



when Pf denotes the greatest prime under Vx. But this formula 

 * See Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 1. p. 67. 



