with Goldbach's Theorem 243 



§ 5. It has been shown by Landau* that 



Sv(/0~„-7r— ^ (10); 



1 ^ ^ 2(logn)2 ^ ^' 



and that Stackel's formula (8) is inconsistent with (10), and ac- 

 cordingly incorrect. 



The same test can be applied to the formula (1) and Sylvester's 

 formula (7). In fact Messrs Hardy and Little wood have shown f 

 that (10) is a consequence of (1) : from which it follows, of course, 

 that the asymptotic formula of the type of (10), furnished by 

 Sylvester's formula, would be in error to the extent of a factor 

 2e~'>'= ri23 ; that Sylvester's formula is therefore also incorrect ; 

 and that if any formula of this type is correct, it must be (1). 



It may seem at first surprising that, in these circumstances, 



Sylvester's formula should give, for fairly large values of n, results 



actually better (as is shown by the results in the table on p. 242) 



than those given by (1). The explanation is to be found in the 



nature of the error term in (1). The modified formula (5), which 



we have already shown to be likely to give better results than (1), 



for moderately large values of n, differs from (1) by a factor of the 



type 9 



1+r^ +.... 

 logn 



This factor does not affect the asymptotic value of v (n), but it 

 makes a great deal of difference within the limits throughout 

 which verification is possible : thus when n= 170,170 it is equal to 

 1"166. When n= 10^", it is equal to 1"087, and its difference from 

 unity is negligible only when n is quite outside the range of 

 computation. It is only such values of n that would reveal the 

 superiority of the unmodified formula (1) over Sylvester's formula. 



1 6. Shortly after the writing of the preceding sections had been 

 completed, Mr Hardy informed us of the existence of a third pro- 

 posed asymptotic formula for v (n), given more recently by V. 

 Brun;|:. The formula to which Brun's argument leads is 



v{n)^2Bn^'~l^^ (11), 



p—z q—2 



where 5= { 1 - " ] ( 1 - " ) ( 1 -^ ) ... ( 1 -r 



" GiHtinger Nachrichten (1900), pp. 177-186. 

 t See their note which follows this paper. 



:!: Archiv for Mathematik (Christiauia), vol. 34, 1917, no. 8. See also § 4 of 

 Hardy and Littlewood's note. 



