Proceedings of the International Congress of Mathematicians
Helsinki, 1978
Recent Work in Additive Prime Number Theory
R. C. Vaughan
I shall concentrate xny attention on matters related to Goldbach's problem, since
this exemplifies the whole of this subject.
Goldbach wrote two letters to Euler in 1742 in which he conjectured (a) that
every integer greater than 2 is the sum of three primes, and (b) that every even
natural number is the sum of two primes. He included unity as a prime, and since
(a) then follows trivially from (b) it is the latter with which Goldbach's name has
become associated. The modern form of this question excludes unity, of course,
and asks that every even integer greater than 2 be the sum of two primes.
It is perhaps of some historical interest that Descartes had earlier made the
same conjecture (essentially), but his comments were only published much later [7].
Not until the early 1920s were any serious contributions made to the subject,
and then there were two important developments, namely the Brun sieve and HardyLittlewood method. These have resulted in three principal lines of attack.
1. Direct applications of sieve methods. There are excellent surveys of earlier work
in Halberstam and Roth [14] and Halberstam and Richert [13]. Let P(x) = Up<xp.
Then the sieve enables us to estimale the number of elements in
{n — p: [n—p9 P(n°)) = 1, p < 77} or {nv. m < n9 (m(n — m), P(n0)) = 1}
provided that 0 is fairly small. It thereby enables us to conclude that if n>n09
then 2n=Pk-\-Pl with k + l*^C9 where P3 denotes a number that is square
free and has at most j prime factors. The most recent result is Chen's theorem
[2], [3] that C=3.
390
R. C. Vaughan
To understand Chen's contribution we must have a little more notation. Let,
for large even n9
£& = {n—p: (n—p, P(n1,1Q)) = 1, n—p square free, p < w}.
Si = i
, 2
, \{bta:p)b}\,
s
2 = ~2 \{b€@- b = pxp2p*9 P i < n 1/3 ^ p2 < p 3 }|,
Both \âi\ and 5^ can be estimated directly by the sieve, which gives a good positive
lower bound for \âi\ — Sx. Elements of & with at least two prime factors in
[rt1/10, «1/3) will make a non-positive contribution to \&\ — Sl9 and no element of
& can be of the form mpxp2pz
with n1/3^p1^p2^p3.
Thus the only positive
contribution to S can arise from those elements which are P a 's.
The expression S2 as it stands is not entirely suitable for an application of the
sieve. However, Chen obtains, by applying the sieve in a novel manner, an upper
bound for S2 which is smaller than the lower bound for \&\ — S1 and thus secures
5 > 0 and so the existence of a p<n for which n—p=P2. In fact he uses the sieve
to estimate
— {{n-p^p*:
PiPzPs < n, n1/10 < Pi < « 1/3 < P2 < jp8. (»-PiPaPai P(w1/2)) = l}|.
This requires, among other things, a new form of the Bombieri-Vinogradov prime
number theorem.
Simpler proofs have been provided by Ding, Pan, Wang [10] and Ross [24].
Graham [12] has shown that the n0 is effectively computable and Ross [25] has
shown that it is possible to restrict the prime variables in various ways.
2. Indirect applications of sieve methods. This stems from Shnirel'man [26], [27]
and was largely superseded by Vinogradov's work (see § 3). However, it is always
useful to have alternative lines of approach to difficult problems. His idea was to
use the sieve to show that
A(x) = \{n ^ x: 2n = px+p2
satisfies
(1)
A{x)?>x
or n = 1}|
( x > 1).
This is readily obtained from upper estimates for
*(»):=
2
1
Pi+Pa=n
provided by the sieve. For instance one may use Cauchy's inequality in the form
(2R(n))2^(2R(n)2)
n^x
n^x
2
n^jc;u(n)>0
1,
Recent Work in Additive Prime Number Theory
391
the left hand side being easily estimated by means of the prime number theorem.
Then one can use theorems about the addition of sets $9 # of natural numbers,
J>-f# = {n:
n
= b + c9 be@9 c£<$ or ne@ or n£<#}
and their Shnirel'man density
inf— \{a ^n: ae^}\,
sé = @9 <g or ^ + #
nel n
combined with (1) to show that there exists a C0 such that for every ws>2 one
has n=Pi + ...-|-/>j with j^C0. The most recent work in this direction has been
due to Klimov, C0 = 6X 109 [18]; Klimov, Pil'tai, Sheptitskaya, C0 = 115 [19];
Deshouillers, C0=75 [8]; Vaughan, C0=27 [32]. This last paper discusses two
variants of the method, in one of which the calculations are easier and give C 0 =27.
The alternative variant, provided certain calculations can be carried out, does
better than this, and thereby Deshouillers [9] has obtained C0=26.
If instead one only asks for a constant C such that every sufficiently large n is
the sum of at most C primes, then the method can be further refined. The most
recent values for C have been C= 10 due to Chechuro, Kuzjashev [1] and Siebert [28]
and C=6 due to Vaughan [31].
Of course, if the calculations could be carried out in the
3. Hardy-Littlewood-Vinogradov method, then we would have C 0 =4. This method
has its genesis in the work of Hardy and Littlewood [15], [16]. They showed, in
particular, on the assumption of the generalized Riemann hypothesis that (a)
*B00 =
2
i
satisfies
and (b)
(3)
satisfies
E(x) = \{n ^ x: 2/7 7* p+p'}\
E(x) = 0E(x1/2+E).
Of course, (2) implies that for n odd and large, n =px +p2 -f-/?3 is soluble in primes
Pl>P2,Pf
Vinogradov [35], by obtaining nontrivial estimates for the sum
a
q-\ (a9 q) = l9 (log N)c < q < N{iog N)~c
p^N
<1
was able to give an unconditional proof of (2). Later Linnik [20], [21] (see also
Chudakov [5]), Montgomery [22] and Vaughan [33] have given different ways of
estimating (4).
(4)
2
e2niap
when
392
R. C. Vaughan
Immediately following Vinogradov's work, Chudakov [4], van der Corput [6]
and Estermann [11] all showed that
0A(x(\ogx)-A).
E(x) =
This was later improved to E(x) = 0(x exp (— c l/log x)) in Vaughan [29], [30]
and then to E(x)=0(xL~ö) in Montgomery and Vaughan [23].
One of the fascinating aspects of additive prime number theory is the continual
interraction between it and other areas of analytic number theory. For instance,
the ideas contained in [33] have recently been used.
(a) to give [34] a new short proof of the Bombieri-Vinogradov prime number
theorem,
(b) by Heath-Brown and Patterson [17] in their recent resolution of the Kummer
problem regarding cubic Gaussian sums (to the effect that the arguments are uniformly
distributed modulo 2n).
Let me explain the underlying ideas of [33]. Many theorems in analytic number
theory depend on estimates for
2Mn)f(n)
(or equivalently 2 f(p))
n
p
iFix}
where f(x) when x£[l9X] is of the form e 9 F real valued, or is a Dirichlet
character x(x)9 and f(x)=0 otherwise. These in turn often depend on estimates
for bilinear forms of the type
(I)
22*mf(mn)
m
and
(II)
n
22*mynKmn).
m
n
The known bounds for these sums are often 'good' when in (I) m is restricted
to an interval[l,w] with u fairly small compared with X and in (II) m is restricted
to an interval [v9 X/v] with v tending to infinity with X.
Vinogradov relates sums over primes to such bilinear forms via the sieve of
Eratosthenes in the form
(5)
/(1)+ _2
1/X<PIêBX
f(P)=
2_
m\P(l/x)
2 Km)f(mn), P(Y) = ft p.
n^X/m
P^Y
The sum on the right is of type I but has the unfortunate defect of including m's
that are close to X. He overcomes this by a combinatorial argument which says
in effect that those m close to X with all their prime factors small occur relatively
rarely and so can be neglected. The terms corresponding to the remaining m can be
rearranged into the form
2 __ 2 xPykf(pk),
V<p^yx
k^Xfp
which is a good bilinear form of type (II). However, in its sharpest form the combinatorial argument is rather complex and not well understood.
Recent Work in Additive Prime Number Theory
393
In [33], this is overcome by using instead the fundamental identity
(6)
So := 2M00/00 = Si-s'.-s'a + s;
n
where
«51= 2 2>("0(log»)/(m");
m Su n
S,=
2 2c,„f(nw),
mSuu n
ss= 2
cm= 2
2»(d)A(k);
dSU k^v
dk=m
Z-tmAWim»),
T»=
2
A*(<0;
Si=2Mn)Kn).
For suitable choices of u and v9 S2 is a good sum of type (I), S3 is a good sum
of type (II) and SA is trivial. Moreover, in Sx the log is easily .removed by partial
summation, whence it also becomes a good sum of type (I).
The proof of the fundamental identity (6) is very simple. It is an immediate
consequence of the observation that
(m = 1),
r l1 (ii
lO (1 m =s£ u).
Thus
SQ-SA + St = 2
III
2' 2*mMn)f{mn)
M>!>
:
2Kd){22Mn)Kdkn)-2Mn)2Kdkri))
dSu
—
ST
k
II
H^O
k
—So.
References
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