[Paper Review] Large Spaces Between the Zeros of the Riemann Zeta-Function

Samir H. Saker|arXiv (Cornell University)|Jun 30, 2009

Analytic Number Theory Research17 references3 citations

TL;DR

This paper establishes new unconditional and conditional lower bounds for gaps between consecutive nontrivial zeros of the Riemann zeta-function using Opial and Wirtinger-type inequalities. Unconditionally, it proves that consecutive zeros often differ by at least 1.9902 times the average spacing—improving prior bounds of 1.9 and 1.9799. Conditionally, under the assumption that higher mixed moments of the Hardy Z-function and its derivative align with random matrix theory predictions, it derives explicit lower bounds for increasingly large gaps, with estimates reaching up to 4.0736 for k=7.

ABSTRACT

In this paper, we will employ the Opial and Wirtinger type inequalities to derive some conditional and unconditional lower bounds for the gaps between the zeros of the Riemann zeta-function. First, we prove (unconditionally) that the consecutive nontrivial zeros often differ by at least 1.9902 times the average spacing. This value improves the value 1.9 due to Mueller and the value 1.9799 due to Montogomery and Odlyzko. Second, on the hypothesis that the 2k-th mixed moments of the Hardy Z-function and its derivative are correctly predicted by random matrix theory, we derive some explicit formulae for the gaps and use them to establish new (conditional) large gaps.

Motivation & Objective

• To derive improved unconditional and conditional lower bounds for the gaps between consecutive nontrivial zeros of the Riemann zeta-function.
• To employ Opial and Wirtinger-type inequalities in the context of the Hardy Z-function and its derivative to establish these bounds.
• To investigate the validity of the conjecture that the maximal gap between zeros, normalized by average spacing, tends to infinity under the Riemann Hypothesis.
• To provide explicit formulae for the lower bounds of the maximal normalized gap Λ(k) under the assumption that the 2k-th mixed moments of Z(t) and Z'(t) match predictions from random matrix theory.

Proposed method

• The paper applies Opial-type inequalities to functions vanishing at endpoints of intervals, establishing a lower bound for the L2k-norm of the derivative in terms of the L2k-norm of the function.
• It uses the Wirtinger-type inequality from Brnetić and Pečarić to derive an alternative lower bound involving the integral I(k), which depends on the function's oscillatory behavior.
• The analysis assumes that the 2k-th mixed moments of the Hardy Z-function and its derivative match those predicted by random matrix theory (RMT), a key conditional assumption.
• By combining these inequalities with asymptotic formulas for the moments of Z(t) and Z'(t), the paper derives explicit lower bounds for the maximal normalized gap Λ(k).
• The bounds are derived by analyzing the inequality chain over intervals between consecutive zeros and taking the limit as T → ∞.
• Numerical values for I(k) and known moment ratios b(0,k)/b(k,k) are used to compute explicit estimates for Λ(k) for k = 3 to 7.

Experimental results

Research questions

• RQ1 What is the best possible unconditional lower bound for the normalized gap between consecutive nontrivial zeros of the Riemann zeta-function?
• RQ2 Can the conjecture that the normalized gap Λ = lim sup r_n tends to infinity be supported under the assumption that the 2k-th mixed moments of the Z-function and its derivative match random matrix theory predictions?
• RQ3 How do Opial and Wirtinger-type inequalities contribute to deriving explicit lower bounds for the maximal gap between zeros?
• RQ4 What explicit numerical lower bounds can be obtained for the maximal normalized gap Λ(k) for k = 3 to 7 under the RMT moment assumption?
• RQ5 Can the structure of the moment ratios b(0,k)/b(k,k) and the integral I(k) be used to determine whether Λ(k) grows without bound as k increases?

Key findings

• The paper establishes an unconditional lower bound of 1.9902 for the normalized gap between consecutive nontrivial zeros of the Riemann zeta-function, improving upon the previous bounds of 1.9 and 1.9799.
• Under the assumption that the 2k-th mixed moments of the Hardy Z-function and its derivative match random matrix theory predictions, the paper derives explicit lower bounds for the maximal normalized gap Λ(k).
• For k = 3, the lower bound is Λ(3) ≥ 2.2265 using the Opial-based method, and Λ(3) ≥ 2.4905 using the Wirtinger-based method.
• For k = 7, the lower bound reaches Λ(7) ≥ 3.7676 using the Opial method and Λ(7) ≥ 4.0736 using the Wirtinger method, indicating a significant improvement in the estimate of large gaps.
• The results suggest that the maximal normalized gap Λ(k) grows with k, supporting the conjecture that λ = ∞ under the Riemann Hypothesis, provided the RMT moment assumption holds.
• The derived bounds depend critically on the values of the moment ratios b(0,k)/b(k,k) and the integral I(k), which are computed numerically for k = 3 to 7.