ON A THEOREM OF HUA

Let p with or without subscripts stands always for prime numbers and E -1(x) = #{n less than or equal to x:n = 3 (mod 24), n not equivalent t o 0 (mod 5) and is not representable as n = p(1)(2) + p(2)(2) + p(3)(2 )}, E-2(x) = #{n less than or equal to x:2\n,n not equivalent to 1 (mo d 3) and is not representable as n = p(1) + p(2)(2)}. It was proved by L.-K. Hua that E-i(x) << x(logx)(-A), i = 1,2 holds for some fixed A > 0. W. Schwarz later showed that any constant A > 0 is permissible. I n this article we establish a short interval version of the above resu lt by proving that (a) E-1(x + H) - E-1(x) << H(logx)(-A) for x(3/4+ep silon) less than or equal to H less than or equal to x, (b) E-2(x + H) - E-2(x) << H(logx)(-A) for x(7/16+epsilon) less than or equal to H l ess than or equal to x, where the implied constants depend at most on A and epsilon.