In the first paper of this series [Proc. Lond. Math. Soc. (3) 72, 261-280 (1996; Zbl 0853.11076)] the first two authors showed that if \(x\) is sufficiently large then the interval \([x-x^{0.535},x]\) will contain at least one prime number. In the present paper the range required is reduced to \([x-x^{0.525},x]\). Since it is currently unrealistic to hope for an exponent better than the value \(1/2\) obtainable under the Riemann Hypothesis, such results are best measured by amount that the exponent exceeds 0.5. In this sense the current paper improves substantially on the previous one, reducing the excess by over 28%.
The basic techniques follow those in the previous paper. One uses a Buchstab decomposition of the sieve function, and attempts to estimate as much as possible asymptotically, using mean value estimates for products of Dirichlet polynomials. Sieve tricks such as ‘rôle reversals’ produce some minor improveents, but the main advantage over the earlier work comes from the deployment of N. Watt’s mean value result [J. Number Theory 53, 179-210 (1995; Zbl 0837.11050)]. This is used in the form \[ \int_{-T}^T \Biggl|\sum_{m\leq M} a_mm^{-1/2-it} \Biggr|^2 \Biggl|\sum_{n\leq N} n^{-1/2-it} \Biggr|^4 dt\ll T^{1+\varepsilon} (1+M^2 T^{-1/2}) \max_m |a_m|^2, \] valid for \(N\ll T\) and any fixed \(\varepsilon>0\). Classically the factor \(1+M^2 T^{-1/2}\) would have to be replaced by \(M\).
A great deal of notation is taken from the previous paper, and the exposition is extremely compact. Thus the reader should not attempt the present paper without a thorough understanding of the principles described in the original article.