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12 G. H. HARDY:
The first is (i-2-s)£(Y), but the second is a new transcendant.
There are ep (Æ) such functions associated with a given k.
The most natural generalisation ofRiemann’s hypothesis is
HYPOTHESIS R. The real part ot a zero ofL(s) does
not exceed -i-.
We do not actually need quite the full force of this
hypothesis, but I shall be content to state it in its simplest and
most striking form.
Our main theorem is then a follows: If hypothesis R
is true, then every large1) odd number is the sum of
three odd primes. The number v3(ri) of
representations is given asymptotically by the formula
____. (27)
where
The product in (27) extends over the odd prime
divisors of n, and that in (28) over all odd primes.
The complete proof of this theorem, and of similar theorems
concerning representations of numbers by four or any larger
number of primes, will appear shortly in the Acta
Mathe-matica. At the moment I shall attempt to explain to you only
(1) how the final formula arises,
(2) why it should be necessary to assume hypothesis R,
(3) why our method succeeds for three or more primes, but
fails for two.
I shall not take these questions in the order in which I
have stated them: I begin with the second. You will remember
that it is necessary to approximate to the function /"(#) in the
neighbourhood of the point h, k. Taking first the simplest
case, suppose that h - ö, k- i, xhtk= i.
We write % = e~lj, and use the well known formula
e-y = -. \y-sT(s)ds (29)
2m y w v ^
of Mellin. From this we deduce
*) That is, every such number from a certain point onwards.
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