A limitation on proving the existence of small gaps between zeta-zeros

Abstract

We assume the Riemann Hypothesis (RH) in this paper. The existence of Landau-Siegel zeros (or the Alternative Hypothesis) implies that there are long ranges where the zeros of the Riemann zeta-function are always spaced no closer than one half of the average spacing. However, numerical evidence strongly agrees with the GUE model where there are a positive proportion of consecutive zeros within any small multiple of the average spacing. Currently, assuming RH, the best result known produces infinitely many consecutive zeros within 0.515396 times the average spacing. This is obtained using the Montgomery-Odlyzko (M-O) method. It is also known that the M-O method fails to prove the existence of consecutive zeros closer than 1/2 times the average spacing. It is a tantalizing hope that the M-O method could still obtain infinitely many consecutive zeros arbitrarily close to 1/2 times the average spacing. We prove however that the M-O method can never find infinitely many consecutive zeros within 0.5042 times the average spacing.

Small gaps and small spacings between zeta zeros

“…Two different proofs of this method are given in [25] or [10] and the limitations of the method are discussed in [20]. This method has also been modified by Conrey, Ghosh, Goldston, Gonek, Heath-Brown [11] in 1985 to produce the positive proportion result µ D ≤ 0.77.…”