On some diophantine inequalities involving primes

0 < 1 * |< [< 5 -* ] 
|£| >
where
N
N
u(k) — k ~ 2d~1(e(kd) — 1) 21 2! e(k(mrj 
n £ — /3)).
ni = 2 n = 2
Now th e last double sum has absolute value at most
A’ 
I 
1
N
J? e(kmrj) 
j 
^ N  || krj
-
i
m  2 
i 
■"
and, by hypothesis, th e num ber on the right is less th a n ^ d2N 2 if 0 < | k | [3“ 2]-
Thus for k in th e la tte r range we have | u(k) | ^ d N 2k ~ 2 and this implies th a t th e first
sum on th e right of (17) does not exceed 
n 2&N2.
o
From the trivial estim ate | u(k) | ^ 2 k ~ 2&~1N 2 we see th a t the second sum on th e
1
right of (17) does n o t exceed 6<5iV2. Thus we obtain | U j < -¡r^N2 and a similar estim ate 
holds for th e absolute value of V. The right hand side of (16) is therefore greater th a n
(/? —
x  — 
d) ( N  — I ) 2 — - t
6 N 2 > j ( p — x — 2 d ) N 2 = S N 2.
However, the left hand side of (16) is clearly less th an F(oc, /3, N) and this gives th e 
required result.

---

Lemma 10. Suppose that N
is an integer greater tha
for 
k =  1, 2 ,. . . , [(4 /i)2]. Then
(18) 
/ I { a xx) 
I { o 2
I {o 3 {sin (n  
log iV)“ 3.
— oo
Proof. For brevity we w rite L(Z, m, n) for l a x + mcr2 + n a 3. From Lemma 8 we 
see th a t th e integral on the left of (18) is given by
N
N
N
2 2 2  {1 — L(Z, /ft, ti)} (log Hog m log ra)*“1.
1 — 2 m — 2 n — 2 
\ L ( l t m , n ) \  ^ 1
This clearly exceeds
i i 2 2 (log Z log m log re)-1 ^ 4 (log V )“ 3ft)(iV),
A i = 
2
m = 2 n = 2 
^
|2 # f» , * ) |£ *
where co(N) denotes th e num ber of integer triplets Z, m, rc with 2 ^ I, m, n <^ N  satis- 
1
fying I L(J, m, «) I ^
2
'.
Now, since || # |I = || — x  ||, all the hypotheses of Lemma 9 are satisfied with 
— <%
= £ = (2/1)-1, V = a 3l | o-j |
and iV replaced by 
K — [(4 /l)-1 jV] + 1. Thus th e num ber of integer triplets I, m, re 
w ith 2 ^ m, n ^ K  satisfying
(19)
- ( 2 / i r 1 <
I — /«(orJ  | 
a
1
1) — 
| a , |) ^ (2 /1 )-1
exceeds (4/1)"1 i f 2. From the left hand inequality of (19) and th e suppositions made at 
the beginning of § 2 it follows th a t I is positive. Similarly the right hand inequality 
implies th a t I does not exceed
• • 
* : £, J 7 
' ;• -• ' 
■' - - 
- - T-
((4 H )-3iV + 1 ) (a3) I a , I"1 + (2 /1 )-1 
+  2/1 + (2 
<
Moreover, since cr3/ | o x | ^ A ~l, there is a t m ost one integer n satisfying (19) for a given 
pair I, m, and thus th e num ber of integer triplets Z, /n, n for which (19) holds with Z = 1 
is a t m ost K . Hence co(N) exceeds
^ A ) - xK 2— K
> i ( 4 / l ) - 3iV2,
and this proves Lemma lO.v 
‘
Lem m a 11. There are positive constants c18, c17 such that
e ( X )
(20)