250 Mr Hardy and Mr Littlewood, Note on 



that Vr (n) is the number of expressions of n as the sum of r primes 

 Then if r is odd we have 



(5-11) v,(n) = o()i>-') 



if V is even, and 



if n is odd, p being an odd prime divisor of ??, and 



(513) B=n[i^^^^^^^. 



where tn- runs through all odd primes. On the other hand, if r is 

 even, we have 



(5-21) Vr(n) = o{n''-^) 



if n is odd, and 



where 



(5-23) c=n|i- 



(^-ir 



if 71 is even. The last formula reduces to (1) of Shah and Wilson's 

 paper when r = 2. 



We have not been able to find a rigorous proof, independent 

 of all unproved hypotheses, of any of these formulae. But we are 

 able to connect them in a most interesting manner with the famous 

 ' Riemann hypothesis ' concerning the zeros of Riemann's function 

 f (5). The Riemann hypothesis may be stated as follows : ^(s) has 

 no zeros whose real part is greater than ^. If this be so, it follows 

 easily that all the zeros of ^(s), other than the trivial zeros s = — 2, 

 s = — 4, ..., lie on the line <t = 'R{s) = ^. It is natural to extend 

 this hypothesis as follows: no one of the functions defined, luhen a- > 1, 

 hy the series 



n" 



possesses zeros luhose real part is greater than ^. We may call this 

 the extended Riemann hypothesis. This being so, what we can prove 

 is this, that if the extended Riemann hypothesis is true, then the 

 formidae (5"11) — (5'23) a?'e true for all values of r greater than 4. 

 The reasons for supposing the extended hypothesis true are 

 of the same nature as those for supposing the hypothesis itself 

 true. It should be observed, however, that it is necessar}", before 

 we generalise the hypothesis, to modify the form in which it is 

 usually stated; for it is not proved (as it is for ^{s) itself) that 

 L{s) can have no real zero between ^ and 1. 



1 



