Messrs Shah and Wilsons paper 249 



and so 



(4-3) f{s) ~ 2^^ (0 -2(1- 2-0 ?(0 - ^1 = ,--2 • 



This is a consequence of our hypothesis : the corresponding 

 consequence of the hj^pothesis (4"1) would be 



(4-31) /(*^>~^- 



On the other hand, it is easy to prove* that 

 (4-4) &)(l) + ft)(2)+ ... +&)(7i)~i«'; 



and from this to deduce that 



<^(.) = 2 



on 



(n) 1 



w* s — 2 



when s—>2. This equation is inconsistent with (4"1) and (4'31), 

 unless (7 = 1. 



It follows that Sylvester's suggested formula is definitely 

 erroneous. 



It is more difficult to make a definite statement about the 

 formula given by Brun. The formula to which his argument 

 naturally leads is Shah and Wilson's formula (12); and this 

 formula, like Sylvester's, is erroneous. But in fact Brun never 

 enunciates this formula explicitly. What he does is rather to 

 advance reasons for supposing that some formula of the type (4"1) 

 is true, and to determine G on the ground of empirical evidence^. 

 The result to which be is led is equivalent to that obtained by 

 taking C= 1-5985/1-3203 = 1-2107 %. The reason for so substantial 

 a discrepancy is in effect that explained in the last section of 

 Shah and Wilson's paper. 



Further results. 



5. The method of § 2 leads to a whole series of results con- 

 cerning the number of decompositions of n into 3, 4, or any number 

 of primes. The results suggested by it are as follows. Suppose 



* Since SA (71) .r™-:; 



as a;-*-l, we have 2w(H)a;"= (SA («) a;"}-~ ,- - -', 



and the desired result follows from Theorem 8 of a paper published by us in 1912 

 (' Tauberian theorems concerning power series and Dirichiet's series whose coefficients 

 are positive', Proc. London Math. Soc, ser. 2, vol. 13, pp. 174-192). This, though 

 the shortest, is by no means the simplest proof. 



The formula (4-4) is substantially equivalent to Landau's formula (10) in Shah 

 and Wilson's paper. 



t Evidence connected not with Goldbach's theorem itself but with a closely 

 related problem concerning pairs of primes differing by 2. See g 7. 



1 1-5985 is Brun's constant, while 1-3203 is 2A. 



