On some diophantine inequalities involving primes

;
(34) 
I 
J
| JJ S  (u;<%) 
j 
{sin (tz 
j
\ J x  
I
( 9 -A ) ( lo g ^ ) f f
^ Ci0iV(log N) 2 
J
I S ( | doc 
0
7j It should be remarked that one of J 2, J 3 can be supposed em pty.
23*

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180
 
B a k e r
, 
On some diophantine inequalities involving primes
i
and by Holder’s inequality the integral on th e right is at m ost ( V 1V 2)2 where
(logxV)f f
V,= J 
\
(7
= 1,2).
0
On substituting | (Xj |  = /?, dividing the range of integration for /? into u n it intervals 
and noting also
/
|S ( /5 ) l2 d/?< 7V ,
0
we deduce easily th a t 
and V 2 do not exceed 2 (log N ) HN. Thus the num ber on the 
left of (34) is a t m ost
2 c10lV2(log N ) ^ h)+H < TV2(log ;V)l(9- A)+i ff < M,
where
J / = /liV 2( lo g iV f s_K
A similar result holds w ith J x replaced by J 2 or / 3 and hence we see th a t 3 M  provides 
an upper bound for the num ber on the left of (34) with J 1 replaced by the interval (33). 
The same inequality is valid w ith S {a joc) replaced by /(We now use Lemmas 10, 11 and 12. By (31) and the definitions of H  and N  we see 
th a t the right hand sides of (20) and (21) do not exceed M . Since also o(N) > (log iV^iV“ 1, 
it follows by dividing the real line into six intervals with end points 0, ± (log N ) hN ~ l, 
± (lo g iV ) H, ± o o and applying the various estim ates obtained earlier for th e corres­
ponding integrals, th a t
J  ! П
S
(tf;*) — Я 7(о-,л)| {sin 
л)/(я«)}2 doc