On some diophantine inequalities involving primes

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On some diophantine inequalities involving primes
By A . Baker at Cambridge
1. Introduction
A web known theorem of D avenport and Heilbronn [2] states th a t if ?.ly . . ., Xb 
are non-zero real num bers, not all of the same sign, with one a t least of the ratios ?ujX} 
irrational, th en for any e >  0 there exist integers x lf . . - , x 5, not all zero, such th a t
The result was originally conjectured by Oppenheim on the basis of a classical theorem
of Meyer to the effect th a t if all the XJXj are rational then th e left hand side of (1) re­
presents zero non-trivially. The Davenport-H eilbronn theorem was later made more 
precise by Birch and D avenport [1] who showed th a t in fact a solution of (1) exists with 
x u  . . 
x 5 all of order s~2~6 for any <5 > 0; thus there exist infinitely m any sets of integers 
x l7 . . 
x- satisfying the inequality
where x denotes the m axim um of the | x t |.
By combining th e m ethod of D avenport and Heilbronn with the H ardy-L ittle- 
wood-Vinogradov techniques familiar in th e study of Goldbach’s problem, it is not 
difficult to obtain an analogue of (1) of the type
where 
denote p rim es1). A n re challenging problem, however, is th a t of de­
ducing a sharper result, analogous to (2), in which e is replaced by a suitable function 
of P i , p 2'>Ps- N either th e original m ethod of D avenport and Heilbronn nor th e later 
modifications as given by Birch and D avenport can be adapted directly for this purpose. 
Our m ain object in the present paper is to introduce a new modification and thereby 
to prove the following
Theorem. Let Aj,A2, A3 be non-zero real numbers, not all of the same sign, with one 
at least of the ratios XJXj irrational. Then for any positive integer n there exist infinitely