Feng and Wu introduced a new general coefficient sequence into Montgomery and Odlyzko's method for exhibiting irregularity in the gaps between consecutive zeros of ζ (s) assuming the Riemann hypothesis. They used a special case of their sequence to improve upon earlier results on the gaps. In this paper we consider a general sequence related to that of Feng and Wu, and introduce a somewhat less general sequence {a n } for which we write the Montgomery-Odlyzko expressions explicitly. As an application, we give the following slight improvement of Feng and Wu's result: infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at most 0.515396 times the average spacing and infinitely often they differ by at least 2.7328 times the average spacing.

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We give an extension of Yoichi Motohashi's theorem saying that if the Riemann zeta-function on the line Re s = 1 attains very small values, then Vinogradov's zero-free region can be improved.

In this note we describe weight functions that exhibit a transitional behavior between weak and strong correlation with the Liouville function. We also describe a binary problem which may be considered as an interpolation between Chowla's conjecture for twopoint correlations of the Möbius function and the twin prime conjecture, in view of recent parity breaking results of K. Matomäki, M. Radziwi l l and T. Tao.

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