ivith Goldbach's Theorem 241 



§4. Goldbach asserted that every even number is the sum of 

 two pnmes, and this unproved proposition is usually called 'Gold- 

 bach s Theorem'. It is evident that the truth of Hardy and 

 Littlewood's formula would imply that of Goldbach's theorem, at 

 any rate for all numbers from a certain point onwards. 



Previous writers, from Cantor onwards, had noted that the 

 HTegularity m the variation of j/(«) depends on the structure of n 

 as a product of pi-imes. In a short abstract in the Proceedings of 

 the London Mathematical Society, Sylvester* suggested the formula 



, . 2n p-2 

 '^'^-logn^p^ (6), 



where, in the product on the right p assumes all prime values from 

 3 to Vw, except those which are factors of n. Sylvester gives but 

 little indication as to how he arrived at the formula and indeed 

 there is much m his paper which is not very clear. It is at once 

 obvious that if 71, n' are two large, but approximately equal even 

 numbers, the values furnished for the ratio v{n):v {n') by formulae 

 (1) and (6) will be the same. For if 



n =2"-p'^ q^ ... 



and n = 2'^ p'^' q'^ 



both formulae will give, as an approximate expression for this ratio 

 the quotient ' 



p-2 q-2'" I p -2 q -2"" 



The actual values of v {n) would however be different. For from 

 formula (6) we should deduce 



\ognp-2q-2-j,'^lji-l- 



Now n ^-2= n Pil:z^ 





'A U (1-- 



p<\'n \ PJ 



where A is the same constant as in formula (1). Also it is known f 



that 



IN 2e-y 



n 1- 



P<sJn\ p) log/i 



^■,1 r ^1°"- f °"^«" ^^««''- ^oc-. vol. 4 (1871), pp. 4-6 (il/atft. Papers, vol. 2, pp 709- 

 711). See also Math. Paj^ers, vol. 4, pp. 734-737. > I'r « c; 



t Landau, Handbuch der Lehre vuii der Verteilung der Primzahlen, p. 140. 



17—2 



