M ÉMOIRES DE LA S. M. F.
S TEPHEN WAINGER
On Vinogradov’s estimate of trigonometric sums and
the Goldbach-Vinogradov theorem
Mémoires de la S. M. F., tome 25 (1971), p. 173-181
<http://www.numdam.org/item?id=MSMF_1971__25__173_0>
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Colloque Th. Nombres [1969, Bordeaux]
Bull. Soc. math. France,
Memoire 25, 19T1, p. 1T3 a l8l
ON VINOGRADOV^ ESTIMATE OF TRIGONCMBTRIE SUMS
AND THE GOLDBACH-VINOGRADOV THEOREM
^y
Stephen
WAINGER
The purpose of this lecture is to give an expository account of the principle
ideas involved in Vinogradov^ estimate for
3^(9) =
i e2^9
p ^n
(in the above summation and throughout
However,
for
S (o)
THEOREM.
p
before we begin to estimate
is a prime).
S ( 9 ) , we indicate why the estimates for
are so important for the proof of the Goldbach - Vinogradov theorem.
(Goldbach-Vinogradov). -
Every sufficiently large odd integer is the sum
of three primes.
We define
r(n)
to be the number of ways to write
n
as a sum. of three primes.
Then the first step of the proof is to observe
1)
To see
r(n) = J^e- 2 " 1116 (3^6 )]3 ^
l), observe
^V 9 "^
I
p^ssn
I
p^<sn
I
e2711^!^-^6
p^n
so interchanging the order of summation and integration, we observe that the right
hand side of
l) , is
I
p^n
The above integral is
1
if
I
I
!
p^n p^ n
1
n = p-,+pp+p^
above just counts the number of ways to write
and the proof of
l)
e-2^11-^!^1^^
0
and
n
as
0
otherwise. So the triple sum
P-i+pp+p^ . This is
is complete.
The idea of (essentially) Hardy and Littlewood is that
r(n) ,
171*
S. WAINGEB
e-2""9^^)]3 d6
r(n)=Jy
+
U
^
V
e-2^9^)^^^.
n
is the union of some very small disjoint intervals which we shall describe later.
What is important for us now is that for every
= a. + -j.
9
6
^th
in
V
(a,q)=.l
,
,
|e| ^ 1
q.
and what is very important
2)
(log n)^ q5£ ——2——.(log n)u
where
u
is some fixed large integer like 100.
Now one may show that
3)
M
>
n
for
n>N
The
and
n
en
2
(log n) 3
odd. (We give some indication of this at the end of our lecture).
main difficulty in the proof is to prove
U)
E
n
=o(__l———) .
(log n) 3
Formulas 3) and 4) of course give the Goldhach-Vinogradov Theorem.
The method of Hardy and Littlewood also dictates the general approach of estimating
\ . Namely
5)
|E | <s sup
n
e^v
|s ( e ) | r
n
"o
By Pavseva^s equality the integral in 5) is
less than
| s _ ( e ) | 2 de
n
7r(n) (the number of primes equal to or
n ). So using the simple estimate
Tr(n)2s
6)
L^ n
'
we have
^ ^-fohr ^ l^9)! n
Notice then to prove h) , we need to show
7)
Of course trivially
[S^(9)| =
o(——1-) .
log n
.
On Vinogradov*s estimate
17-5
l 8 ^ 9 )! ^-f^ .
so we need. only a very modest improvement of the trivial estimate, but this small
improvement was very difficult. Vinogradov^ method proves
THEOREM. - —_—,
If 9
is in
————
the following :
V ^ , _______
then
| S ^ ( 9 ) | ^ c n(log n)^2 {(^ + ^) 1 / 2 + e-0^
8)
for
9 =
Recall that for
9 ^ V
+£
f
?
q.
with
(s-^) = 1 »
l-=a<q
,
n
},
|e| ^ 1 .
n
(log nP ss q ^ ——n——^
(log n)
Thus Vinogradov^ estimate for
|s ( 9 ) | gives 7) and the Goldbach-Vinogradov
Theorem.
We begin by isolating one of Vinogradov1 s basic ideas. Namely in a double sum
^ ^ e2———6 .
I I
n m
there must be a lot of cancellation for most
9 * s , even if the
quite irregular (like say
a = number of divisors of
n
is the MDebius function). To fix ideas consider
9)
s -
?
n=l
with
a,
1
^ e2^
n
and
b
a
and
n
= p(m)
m
b
are
m
where u
,
m=l
(a,q) = 1 . If we neglect the cancellation due to
exp(2i•^•nma•) .,
we may trivially use Schwarz^s inequality to obtain
I s ! ^ n=l
^J m==l
? 1^ ^ A B
where
A = (
^ |a l 2 ) 1 ^ 2
n
n=l
B = (
1 M 2 ) 172
m=l
and
.
However, this estimate may be improved Namely
10)
| s | <;
/q' A B .
We now prove (10) (a la Vinogradov). Using Schwarz^ inequality on the outer sum in
( 9 ) , we see
1T6
S. WAINGER
l
2
^
A 2 I | ^q ^ e 2 ^ ! 2
n=l
in=l
q
q
q
=
A2
=
Now since
A2
i
^
b
F" e2^"^!-^^
•"I ^
^
n=l
-
m- =1
=1
<1
q
m^
m-1
q.
I
i
n^l m^l
b ^
"l^
T e^111^!-"^
n=l
•
( a , q ) = 1 , one sees that the inner sum (which is the sum of a geometric
progression) is
q
if
m
= m?
and
0
otherwise. So we have
|s| 2 ^ q A ^ 2
,
which is (10) .
To prove 8) of course we must also consider irrational
Q^s
but this same
slightly tricky use of Schwarz^ inequality can still be used to obtain cancellation
in that case.
To prove 8) we must use the well known M-obius function
TJ
. We recall
P(l) = 1
y(p^,...,p^) = (-I)11
and
u(n) = 0
for some prime
divides
proof the fundamental property of
II)
R =
p
p^ ^ p_
n
p .
We recall without
Let
if
if
I
d|a
~|[~ p
P ^ /n
.
P(d) =
(
1
if
a = 1
0
if
a > 1
y
that we need :
Then
I e27^9
s^(e) = o ( / H ) +
,^n ^ p ^ n
= o ( /IT) +
^
e
27Tim9
/h <m :sn
(R,m) = 1
=
0( /T) +
I
1 <£m^ n
(R,m) =1
= 0(^n) +
^
1^ m^ n
27rim9
^
p ( d ) e27'11116
d| (R,m)
On Vinogradov's estimate
177
by II). [(x,y) = greatest common divisor of
x
and y]. Interchanging the order of
summation, one sees
S ( 6 ) =0(/n) +
Now setting
Y p(d)
^
e 2 " 1119 .
d]R
1 ^ m^ n
d|m
m = dr , we have
12)
S ( 9 ) =0(/^i) +
V
p(d)
dFR
1
^Tridre ^
!<: r ^ S
a
Now at first glance things may apprear very nice "because the inner sum above is a
geometric progession. However
values of
u(d)
is not zero for more than 0(n)
d , so we are very far from the
proof. What we must do is replace the
double sum in 12) by another double sum with shorter ranges of summation. However,
we are willing to pay a price. Namely we do not need to have such regular coefficies
as
a
= 1 (a
is the coefficient of
use the idea of the proof of
exp 2TTidr9 )
in the sum 12) if we
10).
There are several methods of "shortening" the double sum at the price of introducing rough coefficients. We shall outline here the method given in Prachar
Primzahlverteilung, where the complete details may be found.
Note :
When we use the words relatively easy the reader should not try to stop and
fill in the gaps as they are only really easy for people who are quite familiar
with certain special methods of number theory. For complete
details one should
consult Prachar Primzahlverteilung.
Interchanging the order of summation in 1 2 ) , we have
13)
S . ( 6 ) = 0(/-n) +
I
n
l<s r^ n
I p(d) e2711^ L + F
n
n
d|R
iJd^
where
L =
n
I n— 1
l^e^10^
d|R
^w e2^9
and
F ^ O ^ . I
-^"^r^n
Now
F
I ^e 2 ^ 9
d|R
l<=d<:^
is relatively easy to dispose 6f because in thie double sum above
, , n „ __n
d ^ — ^ —7=s=s
r
vTo
This range of summation on
order of summation in
F
d
is sufficiently small that one may interchange the
and use the fact that the sum on
r
is a geometric
lr 8
^
.
S. WAINGER
progression. Thus we must consider
L
. Instead of considering
^= !<: r^
I e r.—
I
^g
d|r
n
1
d
Thus
G
differs from
L
^
n
L
, we consider
e2
^
1 ^ d <s n/r
has at least one prime factors e
in that now we are not summing over any
whose prime factors are less than
v
e
implies that there are so few such
6 n
°
d^
d*s
<——
°^
all of
. A well known theorem of Rank in
that
x.L.^ Ii, ——c^.
1 <. d ^ n/r
d
e2^106
has all prime factors <-
Thus it suffices to consider G * In G
we now write d as p . j
Q y-i
n
n
____—
prime
e^ g
<-p ^ ^/rT (since p|d , d [ R and R = | [ p ).
P^^T
Define
H
H
is not equal to
d
may be written as
to consider
"-^/^ l^^r. tin
u(j
""iI"
jp . However, it is not too difficult to see that it suffices
Hn
all the sums are much shorter than
n
(since
n .
p > e27 log
n
) ) .
is a triple sum instead of a double sum.
^/To^n^
d (z)
is the number of ways of writing
irregular ; but ^ ( d - ( Z ) ) ^ d ( Z )
estimates
is a
because we have not taken into account the number of ways
Now to change this to a double sum we set
where
p
l^PJ^
G
n
(j^—^
^
e2^10^
H
where
H
n
Note that in
But
n
for
^ [d(Z)|
Z=l
Z = pr , and we have
yiog-n
^"'JlR^
1<
-^•Z
Z = pr .
d- ( Z )
(the number of divisors of
is of course very
Z ) , and very ffood
are known
M
( ^ ^(Z)! 2 )^!^ (log M) 3 } )
Z=l
Notice in the last expression for
H
.
both sums are much shorter than
n
On Vinogradov^ estimate
( J ^
2 ^log i"
since
1V9
n z
<3 ^
and
/
Z
s
e^ 106
n
) . Thus we have achieved our
objective : we have a double sum with shortened ranges of summation and of course
have paid the stated price. If we now break the
Z-sum in
H
into
a
most
log n
dyadic blocs and use Shwarz^ inequality as in the proof of 10) , we may obtain 8) .
At this point you basically understand the idea of the proof of 4). However, we
shall proceed a little farther for completeness. (Note that the trivial estimate
for
H
n
is
| H | <s n
V r-——— d ( Z ) / Z
' nl
Z^e-^^
and this last quantity is
We divide
H
> n
log n ).
into dyadic blocks
K(u)
where
K(u) =
^
d^(Z) . ^
u^ Z^u1
j jR
2irijZQ
(j)
1^4
with
u* <s 2u . We must consider at most log n
log n
such blocks
K(u) . ( A factor of
at this point makes little difference since one may just choose
u
a little
larger in 5) to obtain T) ).
Then
|K(u)|2^
i
|d(z)|2 ^
| ^ p(j)
u<Z<u1
u< Z< u 1 ' J-|R n
1^ J<s Z
^c u(log u)3
^
u<Z<u*
^
j^|R
^
J^IR
^
^l^-Z
. u(Ji)
J^ /
^ ^2^-Z
P(JJ e2^^^
h» \ J ^ /
•s
s c u(log
u(log u
u)j -3
^
I
^
I
n
\
-
^!^ ^
^^
^u
t
^l^u
.cu(logu)3
I
la
If one now observes that for any integer
m9
Aq--^<m<Jlq+^
t^-li
^
I
e^iTiZCj^-jgte
n1
1
are of the form
e2^9!2
I
-
I
u<z<:ul
i^,^^.
^ t ^^ t
Jl , the fractional parts of the numbers
180
with
S. WAINGER
X^
arbitrary and
|e | ^
1
and with each
m , the completion of the proof of 8)
is then
s
a
arising from only one value of
matter of arithmetic.
To make the above observation note that
m 9 = A q 6 + (m-& q)6
=
^
+ J 6
with
- j. < j< -j.
.
Hence we must just observe that
3
(mod 1 )
! f ^ ^ f
for
If
j^ a - j ^ a = q s
for s an integer, the
i •
• i
|j-.-j^|< q so this is impossible.
Ji 5^ J2 •
q|a(j -j ) . But
(a,q) = 1
and
As our main objective was to indicate the proof of 8) (and hence 4 ) , we shall give
only a short summary of 3).
First we shall give the definition of
u =
n
U
j
q ^ (log n)^
(a,^)=l , 0<a< q
j
a,q
a
'^
= {e| l e - ^ l ^ JJ^JL)^}
'
One sees easily that the intervals
Then
u
n
'
q'
J
a,q
n
are disjoint for
n
large.
°
V^ = [0,1]- U^ . The theorem of Dirichilet on approximations of irrationals
by rationals implies that each
6€ V
has the form
n
9 = Q- + £ (log n)^ a^ ^ ^
q
q
n ' q
2
with
I e * | <i\£\<. 1 , (log n)^ q<s — — 2 — — ^
(log n)
To prove 3) we need information on
equal to
n
and (a,q) = 1
as required aboye.
7r(n,q;a) = the number of primes less than or
which are congruent to a modulo g .
The main tool for the estimate 3) is the following.
THEOREM. ^
lk)
(a,q) = 1
w(n
'^
=
^J
uniformly for
1 ^ q <. (log n )
prime to
Here and in the following
q .
, where
^
<(?(q)
&T ^ (n e-0710^)
is the number of integers relatively
c > 0 .
On Vinogradov^ estimate
l8l
We shall not go into the proof of this difficult theorem [See Prachar]. (The
original proof of' the Goldbach-Vinogradov Theorem had much extra complication because the uniformity
q
was not as large as in the above theorem).
The first step in the above proof is the following.
LB4MA. -
For
0^ a <q , (a,q) = 1 , and
15)
1 sSq ^(log n^
,
^ ' ^ ^ ^OCne--/^.
To prove 15) observe
S^) =
^
e2^ =
I e^Pf + I
e21^
p^ n
p<s n
piq
p 11
n
^ma
^
= ^ e 1T1 q 7 r ( n , q , m ) + 0 (log q) .
, m=l
^m,q;=l
Now one finishes the proof of I$7 essentially by using ih) . The next step is the
following.
LEMMA. -
Let
q ^(log n^
, ( a , q ) = 1 , 0^ a < q
and
0 ^ 1 8 1 <s ^ .
Then
16)
^^^U.M S ^ { N ( N 1 ^ 1 ) 0 - ^ 1 ° ]
To prove l 6 ) observe
s
62
.^- <].l
^^-5.-^'
—
One then sums by parts, uses 15) and completes the proof by another summation by
parts.
The remainder of the proof of 3) is now relatively straightforward. One needs
to substitute l6) into the integral for
U.
and keep tract
of the error terms.
Supported by an N.S.F. grant at Faculte de Sciences at Or say.
University of Wisconsin
Department of Mathematics
Madis9n
Wisconsin ( U . S . A . )