A Note on Gaps between Zeros of the Zeta Function

Abstract

Recently Montgomery and Odlyzko [1] showed, assuming the Riemann Hypothesis, that infinitely often consecutive zeros of the zeta function differ by at least 1.9799 times the average spacing and infinitely often they differ by at most 0.5179 times the average spacing. We improve their result by a better choice of a certain weight function which occurs in their proof and we give explicit bounds for what is possible by this method. In addition, we indicate a method of proof which is considerably simpler than that of Montgomery and Odlyzko.LetIn logy 271 where y ^ y' are consecutive ordinates of zeros of ((s). Note that the average of y' -y is 27r/log y.THEOREM. If the Riemann Hypothesis is true, then k > 2.337 and /i < 0.5172.We indicate the modifications in [1] that lead to our theorem. Let n. k nk $ K where 2sin(7rc(logn)/logr) = n log n so that |gf(n)| ^ --and K = V~A for some small 6 > 0. By (20) and (21) of [1], if logT h(c) < 1 for some choice of c, a ki and S > 0, then A ^ c, and if h(c) > 1 for some choice of c, a k , and S > 0, then ix ^ c.To obtain the bound for k we choose a k = d r (k)k~1 12 where d r is multiplicative and for a prime p,

A note on the gaps between consecutive zeros of the Riemann zeta-function

“…In fact, it is known that this is not attainable using Montgomery and Odlyzko's method with Dirichlet polynomials of length ≤ T . Specifically, in [2], it is shown that h(c) < 1 if c < We have not been able to prove that our bounds for λ and μ in Theorem 1.1 are the optimal bounds for our choice of coefficients {a ± k } in (2.2). In the special case of r = 1, this optimization problem has been solved (in terms of prolate spheroidal wave functions).…”

“…Very little is known unconditionally; however, Selberg (unpublished, but announced in [12]) has shown that μ < 1 < λ. Assuming the Riemann Hypothesis, numerous authors [2,5,7,8,10] have obtained explicit bounds for μ and λ. Theorem 1.1 improves the previously best known results under this assumption which were μ < 0.5172 due to Conrey, Ghosh, and Gonek [2] and λ > 2.6306 due to R. R. Hall [5]. The results in Hall's paper are actually unconditional, but a lower bound for λ can only be obtained if the Riemann Hypothesis is assumed.…”

“…In [8], by an argument using the GuinandWeil explicit formula for the zeros of ζ(s), Montgomery and Conrey, Ghosh, and Gonek [2], expanding on an idea of Mueller [10], have given an alternate and much simpler way of viewing this problem. Let…”

Notes on pair correlation of zeros and prime numbers

“…By a different method (on RH), Montgomery and Odlyzko [30] improved on this result and obtained the upper bound 0.5179. Conrey, Ghosh, and Gonek [5] later replaced this by 0.5172.…”

On r‐gaps between zeros of the Riemann zeta‐function

“…To prove our theorem, we adapt a method developed by Conrey, Ghosh, and Gonek [3] on gaps between consecutive nontrivial zeros of ζ(s) in the interval [0, T ] for T large. The method is conditional on the Riemann Hypothesis.…”