252 My^ Hardy and Mr Littlewood, Note on 



The numbers of pairs differing by 4, which should be roughly the 

 same as those of pairs differing by 2, are 



9, 26, 41, 63, 86, 107, 121. 



7, 



Brun, ni his note ah'eady referred to, recognises the corre- 

 spondence between the problem of §§ 2—4 and that of the prime- 

 pairs differing by 2, and realises the identity of the constants in- 

 volved m the formulae ; but does not allude to the more o-eneral 

 problem of prime-pairs differing by k. He does not determme the 

 fundamental constant A, attempting only to approximate to it 

 empirically by means of a count of prime-pairs differing by 2 and 

 less than 100000, made by Glaisher in 1878*. The value of the 

 constant thus obtained is, as was pointed out in § 4, seriously in 

 error. The truth is that when we pass from (6-1), which, when 

 k = 2, takes the form 



2 A{v)A{v + 2)r^^An, 



to (6-3), the formula which presents itself most naturally is not 

 (6-3) but "^ 



(7-1) i\r3(n)o.2yir--^. 



J (log^)- 



This formula is of course, in the long run, equivalent to (6-3) 

 But 



(log xf (log ny \ ^ log n "^ (log nf "^ " ' 7 ' 



and the second factor on the right-hand side is, for n = 100000 far 

 from negligible. Thus (6-3) may be expected, for such values of 

 n, to give results considerably too small. 



}^C^^ *^^® *^® ^^^^®^^ ^"^^^* ^^ integration in (7-1) to be 2 we 

 find that the value of the right-hand side for n = 100000 is to' the 

 nearest integer, 1249, whereas the actual value of i\^., (92) is, accord- 

 ing to Glaisher, 1224^ The ratio is 1-02, and the agreement seems 

 to be as good as can reasonably be expected. 



Tir "Sf ^^l^^^ation of prime-pairs has been carried further by 

 Mrsfetreatteild, whose results are exhibited m the following table: 



*'ri '^^■- • ^^e number of pairs below 100000 is 1225 



t iiie series is naturally divergent, and must be closed, after a finite number of 

 terms with an error term of lower order than the last term retained 



^ Glaisher reckons 1 as a prime and (1, 3) as a prime-pair, making 1225 in all. 



