On some diophantine inequalities involving primes

 
| 
b 
| (log N ) Hy
which are obviously valid if | fi j <; (log N ^ N * 1.
Lemma 2. Suppose that N  > (16 B ) A (log A )4H. Then
i
(7)
/
| / ( M ) 7 ( M ) /(M >
(log A T 2*.
(log 
N ) H y - i
Proof. By Abel’s lemma we have, for all real <%,
(
8
)
2( | ^ min (4iY/log N, |i « H"1).
Thus for /? in the interval (log N ) HN  1 iS /9 iS (2 
B  1 we obtain | /(¿>,/9) | < /? ’. Hence 
th e contribution of this interval to the num ber on the left of (7) is a t m ost
Q
O
J f}~
3
d(i
= -t 7V2(log iV)*
(log 
N ) “ X '
Hx- l
3) 
This can be written in the alternative form p(k)g>(q)l(p(k), where k = ql(m^q); see G. H. Hardy and 
E. M . Wright, An introduction to the theory of numbers (Oxford, 1960), Theorem 272, p. 238.
s d e i r n j

---

1
We now consider th e interval (2 B) 1 ß 
For each such ß there exists a
i 
* 
_i
pair of relatively prime integers a, q w ith 1 <1 a ^ q ^ N 2 such th a t | qß — a | < N K 
Suppose first th a t q > B. Then q cannot divide bj and thus abj = k }q + lj for / = 1, 2
, 
3, 
where k h lf are integers w ith 1 
< q. It follows th a t ¿>¿/8— k j = || 
|| > (2 g )-1;
for clearly 
r
— k j — Ijlq + (bjß -
kj -
Ij/^ g - ^ l - BN~%) >  (2
1
and similarly 1 — (b}ß — k ß > (2q)~\ Since q ^ N 2, we see th a t (8) implies th a t the 
integrand on th e left of (7) is a t most
(2iV2)3 ^ i . V 2(logiV )-2".
1
The p a th of integration has length a t m ost 
and thu s the num ber on the right exceeds
the contribution to the integral of those ß for which q > B.
Now suppose th a t q 
B. Then noting th a t (61? 62, bß = 1, we obtain as above 
| I (b}ß) | < 2q <; 2 B  for one at least of th e integers / = 1, 2, 3. Thus the integrand on 
th e left of (7) does not exceed 2 B N 2. F urther, the path of integration over those ß for 
which q 
jB has length a t most
2 
2
2 q~1N~*<. 2 B N ~ i
l^q^B a — 1
Combining our results we see th a t th e num ber on th e left of (7) does not exceed
- |iV 2(logiV )-2" + 4 
B *
№
< iV2(log iV)~2H,
and this proves Lem m a 2.
Lem m a 3. Suppose that
(9) 
{ (8 //)-4log i V p > B .
1
Let m be art integer such that \ m \ <^-prNB K Then
‘ d ^j 
- ; j
(10) I j e(— 
> ( 6 B 2) - 2N 2(log N ) ~3
| 3R 
j
where c5 > 0 depends only on H.
' ~
 
; : , 
‘ 
■
 - ■
_ 
j 
* - 
• ' 
j 
j
Proof. We note first th a t the hypothesis of Lemma 2 is satisfied; for we have
N  = e>°*y > ((877)!)-1 (log 
N ) s"  and (log iV)4H' > (8
For * on 
4 write
U(x) =
e(— 
oi m) 
¡J S(bj*), 
V ( c x )  =
CQ(bß (1 
; = 1
Jou rn al 
für 
M ath em atik . Bd. 22.S 
22
B a k e r , On some diophantine inequalities involving primes 
169

---

where /} = oc — a/q. From Lem ma 1, noting th a t Cq(bp 5S min (
also (8) we obtain
| 
U(oc) — V(oc) | < c6(m in (iV/log N ,  | 0 I"1))2
where c6 depends only on H. The integral of the squared factor on the right over 
| p | < (log N ) HN ~ l, indeed over all p from — oo to oo, does not exceed 4iV/log N. Hence
170
 
B a k e r , On some diophantine inequalities involving primes
1
f {U(x) — 7(*)} doc
II a = 1 
J
(«*, 0 - i matq
< c7B N 2(log N) ~H,
where c7 depends only on H . We consider the integral over V (th a t, from Lem ma 2, the error introduced by replacing th e range of integration 2Jl0 q 
1 
1
with the interval — 
p  
is less th an
A 
A
i
{ B { y ( q ) ) - ' N 3(log iV)-2H} ^ 2 £ iV 2(log N Y 11.
Thus th e num ber on th e left of (10) differs from
(11) 
M m ) | 
2  
f J 7 C ,( ^ ) )
l^q^(logX)H 0 = 1
by Jess th an c8B N 2(log N) ~H, where
co(m) = j e(— Pm) I I I(bj P)dp
and c8 depends only On H.
Clearly v(m) > co(m) > (log iV)“ 3 r(m ), where v(m) denotes th e num ber of integers 
mx, m2, m3 between 2 and N  inclusive satisfying th e equation
b1m 1 + b2m 2 + b3m 2 = m.
Since v(m) is trivially less th an N 2 we see th a t the error introduced by summing over all 
positive integers q in (11) is less th an
N 2B 3
2
(<
p
(q
))~
2< c9 -V2 
(log 
JV )"K
v 
. 
q > ( \ o g X ) H
i'
The num ber on the left of (10) therefore differs from a>(m)(&(m) by at m ost the second 
term on the right of (10).
I t rem ains only to prove th a t v(m) > ( 6 B 2)~2N 2. L et b = ( \ b1 \J b2) so th a t 
&3) = 1. Since N  > (16jB)4 it is easily verified th a t there are a t least (6 B 2)~~1N  
integers k w ith
— 2 N ( 3 B ) ~ 1 ^ k ^ — N ( 3 B ) - 1 — 2 B
I
such th a t kb = m (m o d & 3). By virtue of th e hypothesis \ m \ < ~ ^ N jB“ 1, the integer
o
m 2 for which kb J- m 2b3 = m clearly satisfies 2 <[ m3 
N.  Similarly for each k there
4) Verified as in the proof of Lemma 7; see also earlier footnote.

---

are at least ( 6 B 2)~l N  integers m l such th a t
2 N ( 3 B ) ~ 1 + 2
N B ~ l
and m 1b1 + m 2b2 = kb, where m 2 is an integer satisfying 2 
This gives the
required estim ate for v(m) and completes the proof of th e lemma.
Lemma 4. Suppose that h > 1. Let oc be a real number for which relatively prime 
integers a, q exist such that
(12) 
| 
qx — a |
< (log N ) hfN, (log N ) h N j (log
O-A)
Then both 
S(x) and I (oc) have absolute values not exceeding c10N (log N) 2 , where c10 
depends only on h.
Proof. The result for 
S(x) is due to Vinogradov; for a proof see, for example, P rachar 
[3], Satz 6. 1, p. 189. Our estim ate follows im m ediately from this on noting th a t
(q-i +
qiy-i) l < 2 (log 
N )~^and 
< A !(lo g iV ft
To obtain th e result for 
I (x) we use th e estim ate | 
| 
|| 
[|-1 given by (8).
h
The inequalities (12) imply th a t | | * | | > (log N) ~N~l provided N  exceeds a suitable 
num ber depending only on h\ for otherwise the nearest integer b to oc would satisfy
\ qb — a \  <: q( log N ) * N - X + (log N ) hN ^  ^ (log N)~* + (log N ) hN ~ \
The num ber on th e right tends to zero as N  tends to infinity bu t th a t on the left is a non­
zero integer and th us a t least 1. Hence the required inequality for || oc || holds if N  is 
sufficiently large. For the rem aining N  we use the trivial estim ate | I(oc) | N  and the 
lemma follows easily.
Lem ma 5, Suppose that (9) holds. Then for all oc on m we have
(36- H )
(13) 
! S ( b}oc) I < cn B N (log N) 
8 
(/ =
2, 3),
where cn depends only on H.
Proof. We write
S ( x , k ) = £
e(ocp)
' 
v 
p  prim e
\ 
■: •
so th a t S(oc, N) = iS'fa), It suffices to prove (13) for / = 1 since the proof for j =  2 or 
3 is similar.
If oc is on m there exist relatively prime integers a, q such th a t (12) holds w ith
h = H. Let d == (q, | bx |) and let q = dqx, a | bx | = dax so th a t 
(a1, qx) = 1. Since
d ^ I 
I ^ B  < (log N ) 2 > we have (log N ) 2 
< qx < N /(log N ) 2 , and th us if 
k is
JL
1L 
1L
any integer satisfying N /[log N ) 4 th en (log k ) 4 < qx < k /(log k ) 4 . Hence
1
applying Lem ma 4 w ith h, oc, N, a, q replaced by ~^H, a bj q, k, ax, q1 respectively we 
obtain
( 3 6 - /7 )
(3 6 -7 7 )
| S ( a b j q , k) \ < c12k (log k) 
8 
< c12iV(log N) 
* 
,
B a k e r , On some diophantine inequalities involving primes 
171
22
*

---

//
where c12 depends only on H. If A; is a positive integer not exceeding AT/(log iV )4 the 
result holds trivially.
Now from the equation
S ( b xa) = * 2 S ( a b J q , k) {e(bxfik) - e ( b j \ k  + 1))} + S (ab1lq)e(bi p N ) y 
where /? = a — a/gr, it follows th a t
(3 6 - H )
I S {bt*) | < c13N (log N) 
* 
( N B  | p | + 1) 
where c13 depends only on / / , and this gives (13) since N  | /3 | < 1.
1
Lemma 6. Suppose that m is an integer such that | m | ^ B and @(m) > —. Then
Ji
for any d > 0 there is a number c14, depending only on d, such that (5) possesses a solution 
in primes p x, p 2, p 2 each less than cf*.
Proof. We begin by defining H  = 120 ([¿_1] + 1). Let c15 = c5 + cn +  1, where 
c5 and cn  denote the num bers which appear in Lemmas 3 and 5. Then c15 depends only 
on <5. Define N  to be th e least positive integer satisfying
(14) 
{(8/7)~4 log iV}^ > c15B 7.
Clearly N  is less th an c**, where c14 is a num ber depending only on d. F urther, (14) 
implies th a t N > H >  100 and
11-2
log N > (8H )AB H > 2 log (3B) ^ log ( 3 B  | m |).
Thus th e hypotheses of Lemma 3 are satisfied. We proceed to prove th a t (5) possesses 
a Solution w ith primes p ^ p ^ P ^  not exceeding N.
L et U [a) be defined as in the proof of Lemma 3. The num ber of solutions of (5) 
in primes p Xl p 2, p 3 not exceeding N  is clearly given by I  -f / , where
I = f U(a)da, 
J = f U(a)da.
W 
m
From Lem m a 5 we see th a t