On some diophantine inequalities involving primes

- e ( X )
where
B a k e r, On some 
diophantine inequalities involving primes
 
175
Q{JS)
r  
I 
3 
3
1 I n S(cTjX) TI I(Oj
0
t) {sin (ji < c16 /L V (e (iV ))-\
J 
\ J —
1 
) —1
q
( N ) =
iV-V’(l0g*v)l.
■
, 
* 
• 
' ■
• I ;
Proof. As in th e proof of Lem ma 1 we deduce, by means of a refinem ent of the 
prim e-num ber theorem , the existence of positive constants c17, c18 such th a t

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176
Пакет, On some diophantine inequalities involving primes
for / = 1, 2, 3 and all real 
oc.
Thus if | 
oc
| 
q{N) we have
| 
S(ajtx) — 7 (a #<%) I <
Then writing the difference of the products in the integrand on the left of (20) as the 
sum of seven term s each including at least one factor of th e type 
S { a j oc)
— 
I ( a ¿ a ) ,
and 
using also the trivial estim ates 
j S ( a j oc)
| 
N,
|/(N
and | sin 
(noc)
| ¿J 
7i
| 
oc
|, 
we see th a t the num ber on th e left of (20) does not exceed
. 
<**)
2 8 c l9A N z e - 2r"«0*y)' f doc = 5 6
c
19A N (
q
( N ) ) - \
- e ( V )
This proves the lemma.
Lemma 12. We have
00
(21) 
f
| 
77
 
S (
o p )
! 
{sin { 7 i * ) l ( j t x ) } 2 
doc 
<  
log N) ~H,
J  
I J ~ 1 
I
( lo « A ) H
and the same holds with S ( o }oc) replaced by I(Gjoc).
Proof. From Holder’s inequality and th e trivial estim ate | S ( a 3oc) | i
th a t the integral on the left of (21) does not exceed 
N { U 1 U 2)2
, where
U j
=
/
I 
S { a } oc)
j
2
{sin 
( jz o c )I{ tio c
) } 2
doc
(/ = 1, 2).
( lo g X ) ff
On substituting /3 = | Oj [ 
oc
and dividing th e range of integration for /3 into unit intervals 
we obtain 
^
h
 4* 1
I U , \ £ 2
n = J
r, | 
(nn)~2 j \S(fl) \2dp
where J — [(log N ) H i Gf |], The integral on th e right is merely the num ber of primes 
and thus
00 
f
A
. 
Q
; :
\ U f \ ^ 71- 2 1 <7 j I N  J 7T2 <
2 1 a j I N
f x ~ 2 dx.
n ~J 
X 
J
J - l
M i ,
Since | 
g
j [ {J —  i) 1 < 2(log.V) 
the required inequality (21) clearly follows.
The corresponding result w ith S(GjOc) replaced by I{G]
0
c) follows similarly, on 
noting th a t for each positive integer n
« 4 -1
f  | 7 (/?) 
\2 dp — (log 
m)~2
^ N. 
J
m
 — 
2
.
.
.
 
n
■
'* l :u - 
- f \ 
1-
Lemma 13. Suppose that h is an integer > 1 , that 0 <
^ (log 
and that
(log N ) h >  (10A) LI. Suppose also that integers a1, a2, qu q2 exist such that for j  = 1 , 2,
a,#= 0, 1 ^ q, ^ (log N ) h and
(22) 
I ? ja ;a — a ; ! < (log

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Then there are relatively prime integers a, q with 1 
q 
(log iV)4/I satisfying the inequality
(23) 
< N ~ {
Proof. There exist, as is well known, relatively prime integers a, g such th a t (23)
i
holds and 
N*. We proceed to prove th a t, by virtue of the above hypotheses,
the stronger inequality q <] (log JV)4* is in fact satisfied.
We note first th a t (23) can be w ritten in the form
I ?2q
^M
q
M (
a/i) i < I 
I
It then follows easily from (22) th a t
B a k e r , On some diophantine inequalities involving primes 
177
(24)
^.172^1*1 | (log 
N)h
1 
TV
q
N 2
Now ajq does not exceed 2 A  and both and q2 are at m ost (log N ) h. Hence the num ber 
on the right of (24) does not exceed
A  (log N ) 2h ( q N ^ y 1 + (log 
+ 2/1 (log
and this is less th an
(log N ) 3h ( q d y 1 + 3(log 
N ) 3^ - 1 < A (log N ) - H ( q - 1 +
5) < ?-»(l0g
1
by virtue of the inequalities A  
<
(log 
iV )*, 
N  
>
16 (log N ) sh and q ^ N 2.
On the other hand, the num ber on th e left of (24) is rational. Since th e deno­
m inator cannot exceed g(log7V)Ä it follows th a t its only possible value is 0. This, 
together w ith the fact th a t (a, q) —- 1. implies th a t q divides axq2, and from (22) we obtain
axq2 < {A(\og N ) 2h + . (log N ^ N ' 1} (log N ) h < (log N ) Ah.
Thus Lem m a 13 is proved.
Lemma 14. Suppose that 
< 0, X2 > 0, A3 > 0. Suppose also that there exist £ > 0 
and infinitely many sets of integers P 2, P 3, Q2, Q3 such that for j = 2, 3, Qj > 0, (P ^ Qj) = 1 
and
(25) 
\ Q f ( X p i) - P j \ < e - * ,
where Q denotes the maximum o' Q2, Q3. Then for any positive integer n there exist infinitely 
many primes Jp1,/? 2, p 3 satisfying (4).
Proof. L et d — (Q2Q3y P 2Q3, P 3Q2). Since ( P j y Qj) =  1 it follows easily th a t 
d(Q2, Q3). We p u t
(26) 
V- bi = — Q i Q * d r \ b2 = - P 2Q3d ~ \ b3 = - P 3Q2d ~ \
and note th a t 617 i>2, b3 satisfy all th e conditions required a t th e beginning of § 2 provided 
Q is sufficiently large. F urther, | b1 1, b2, b? are all clearly less th a n B  =
c
2QQ2, where 
c20 depends only on A1? A2, X3.
We now suppose th a t n is a positive integer and apply Lem ma 6 w ith
1
d = -^m in (£, 1 /n). This, together with Lemma 7, shows th a t there exist primes p t1 p 2, p$
Jou rn a l 
für 
M athem atik. B d . 228 
23

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such th a t (5) holds w ith m — 1 or m — 2 and the m aximum p  of p XJ p 2, /?3 is less th an
c ^ , where c21 depends only on n and £. Thus we have
(27) 
( lo g /)) 215 < c 22(>,
where c22 depends only on n, C, Alt A2, A3. From (26) we see th a t
(28) 
+ A2/>2 + A3p3
= 
— + 
637,2 + 
b3Pi)'
Hence, using (25), (27) and noting th a t 1/(2d) = 2 m ax (1/C, n ), we obtain
(29) | Aj/?! -f- A2p 2 “1“ A3jp3 |
< 2 | Aj | 
+ 2 | Aj | d(+ c24(log p ) ~2n,
where c23 > 0, c24 > 0 depend only on n , f, An A2, A3. Now (? assumes arbitrarily large 
values and this clearly implies the same for p except possibly in the case th a t the num ber 
on the extrem e left of (29) takes the value 0 for all b ut finitely m any of the sets p x, p 2, P
3
-
In the la tte r case, however, (28) implies th a t
2p  max (| (?2(A2/Aj) —
P 2|
, | |) ^
^ 1
and (25) shows th a t p  again takes infinitely m any values. Finally (4) follows easily from
(29) provided p  is sufficiently large, and this proves the lemma.
178
 Ba k e r
, 
On some diophantine inequalities involving primes
4. Proof of Theorem
We suppose, as we m ay w ithout loss of generality, th a t Xx < 0 , A2 > 0, A3 > 0 
and XJX2 is irrational. Let n be any positive integer. We assume th a t all the primes 
jPi,J92? /?3 occurring in th e possible solutions of (4) do not exceed a positive integer c25 
and we shall deduce a contradiction.
L et H =  10n and p u t h =  4 H. We denote by c10, c16 and c17 the constants appear­
ing in Lemmas 4 and 11 and suppose, again w ithout loss of generality, th a t c10 > 1, 
c16 > 1 and c17 < 1. From Lem ma 14 we deduce the existence of a num ber c26 de­
pending only on /i, A1? A2, A3, w ith 0 < c2Q < 1, possessing the following property. For 
all integers P 2, P 3, Q2 > 0, Qz > 0 w ith (jP2, Q2) = 1 and ( P 3, Q3) =  1, the inequality
(30) 
\ Q , ( W - P , \
> c Me-*C
holds either w ith / = 2 or w ith j =  3, where C = 1/(8 h) and Q denotes th e m axim um
of Q2 and Now let 
pi 
and 
v 
denote the m axim um and minimum of 1, | 
Xx 
|, A2, A3 respectively 
and let e be any positive num ber < v satisfying
(31) 
e " 1 > /(4 0 7 7 ) ! {c10c1Qc25c~7l
.
Define N  to be the least positive integer such th a t (2 log N ) H > e“”10. Then clearly 
N  > 100 and (log N ) H < e~10. We proceed to establish th e existence of primes p x, p 2, p 2
•) N ote th at (30) holds with 
j 
=  2 if A3/Ax is rational.

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B a k e r, On some diophantine inequalities involving primes
179
such th a t (3) holds and for which the m axim um p  satisfies c25 < p  ^ N. This will suffice 
to prove th e theorem , for we then have
e < (log N ) - ^  = (log N ) —  ^ (log p) ~n
and this contradicts our original supposition th a t 
(
4
) 
possesses no solution with p > c25.
We define now a = Àje~l for j =  1, 2, 3 and p ut A — jus''1. Then, since e < v < //, 
it follows th a t oq, c 2, or3 satisfy all the conditions required a t the beginning of § 2. F urther, 
from (31) we obtain
(32) 
(2 log N ) H X T 10 > A * e ~ 1 > ( l O h ) \ A
and thus A  satisfies the hypothesis required in Lemma 13. Consider now the hypothesis 
of Lem ma 10. If N  does not satisfy the required condition then, on noting th a t 
g
jlo’ 1 = Aj/Aj, N >  (2 log N ) H and using also (31) and (32), we obtain
W k X M  < ( 4 A Ÿ N - 1 <
c
2, N - \
where k denotes a positive integer 
(4/ l ) 2. Since, again from (31) and (32),
1 
n 
( \ 
\ 8 h
(4/1)2 < 16eï (2 log N ) T  <
log N \
it follows th a t there exist relatively prime integers P 2, Q2 >  0 which do not satisfy (30)
w ith / = 2, C = 1/(8h) and Q = [(4/1)2]. From th e analogous result w ith A3 in place of
A2 we see th a t, by interchanging o,2/(t1 and a j a 1 if necessary, it m ay be assumed, w ithout
1
loss of generality, th a t the hypothesis of Lem ma 10 is valid. Moreover, since N* > c^1, 
we deduce in a similar m anner th a t there cannot be integers P 2J P 3, Q2, Q3 such th a t, 
for j  = 1 and 2, 1 
0 °g N ) Ah, (P Q j ) =  1 and
I Q i W o i ) - P i  I <
Thus if the conclusion of Lem ma 13 is valid, then it will not hold w ith o’2/u 1 replaced 
by o J o x.
For any real oc there exist integers a 17 a 2, a 3, qt , g2, q3 such th a t for / = 1, 2, 3 
N I ( l o g N )h, (22) holds and either (a;., qj) = 1 or a} = 0. The latte r alternative 
implies th a t \oc\ <  (log N ) hN~K From Lemma 13 and the analogous result w ith place of (j2 we see th a t if the first alternative holds and 0 < oc ^ (log N ) h then one at
least of th e integers qx, q2, q3 exceeds (log N ) \ I t follows from Lemma 4 th a t the interval
>; 
. 
M 
my-- 
■
■
' 
. . . . . .
:
(33) -
. 
- u
(log N ) hI N ^
^ (log N ) H
can be divided into three disjoint m easurable subsets J ly J 2, J 3 such th a t | S ( a joc) | does
jM )
not exceed c10JV(logiV) 2 
on J j for / = 1, 2, 3 7). Then clearly