[Paper Review] On the roots of the equation $Z'(t)=0$

Jan Moser|arXiv (Cornell University)|Mar 5, 2013

Analytic Number Theory Research6 references3 citations

TL;DR

This paper establishes that the Lindel"of hypothesis implies a significant contraction in the spacing between consecutive odd-order zeros of $ Z'(t) $, demonstrating that such zeros—corresponding to local maxima or minima of the Riemann zeta function on the critical line—occur densely within intervals of the form $ (T, T + T^{\epsilon} \psi(T)) $, where $ \psi(T) $ is any slowly diverging function. The result improves upon prior exponent bounds in zero-spacing estimates by leveraging trigonometric sum estimates under the Lindel"of hypothesis.

ABSTRACT

We have proved in this paper that the Lindel\" of hypothesis generates essential contraction of distances between consecutive odd-order zeros of the function $Z'(t)$. This paper is the translation of the paper \cite{11} into the English except part 8 that we added in order to point out the I. M. Vinogradov' scepticism on possibilities of the method of trigonometric sums.

Motivation & Objective

• To investigate the distribution of odd-order zeros of $ Z'(t) $, which correspond to local extrema of $ \zeta(1/2 + it) $, under the assumption of the Lindel"of hypothesis.
• To improve upon existing exponent bounds in the spacing between consecutive zeros of $ Z'(t) $, particularly refining the classical Hardy-Littlewood exponent.
• To assess the limitations of the trigonometric sum method in estimating the Hardy-Littlewood integral $ \int_0^T |\zeta(1/2 + it)| dt $, in light of I.M. Vinogradov's scepticism.
• To provide a new asymptotic representation of the Hardy-Littlewood integral with a negligible error term, independent of trigonometric sum estimates.

Proposed method

• Derives an explicit asymptotic expansion for $ Z'(t) $ using the functional equation and contour integral representations, leading to Formula 1: $ Z'(t) = -2\sum_{n \leq \sqrt{t/2\pi}} \frac{1}{\sqrt{n}} (\vartheta'(t) - \ln n) \sin(\vartheta(t) - t\ln n) + \mathcal{O}(t^{-1/4}\ln t) $.
• Transforms the derivative expression into a smoothed sum via a cutoff at $ P_0 = \sqrt{T/(2\pi)} $, yielding Formula 2 with logarithmic weights $ \ln(P_0/n) $, improving convergence.
• Introduces a sequence $ \tilde{t}_\nu $ defined by $ \vartheta(\tilde{t}_\nu) = \pi\nu + \pi/2 $, which aligns with the phase of the sine terms and allows for summation over structured intervals.
• Applies the hypothesis $ |S(a,b)| < A(\Delta)\sqrt{a} t^{\Delta} $ for $ 0 < \Delta < 1/6 $ to bound partial sums of $ Z'(t) $ over intervals $ (T, T+H) $, leading to estimates (2.4) and (2.5).
• Uses alternating sum techniques to separate even and odd indexed $ \tilde{t}_\nu $, deriving Lemma 3 which shows a sign-reversing imbalance in $ Z'(\tilde{t}_\nu) $, implying sign changes and hence roots of $ Z'(t)=0 $.
• Contrasts classical trigonometric sum methods with a new asymptotic representation of the Hardy-Littlewood integral (A.6), showing a negligible error $ \mathcal{O}(\ln T / T) $, independent of sum estimation.

Experimental results

Research questions

• RQ1 Does the Lindel"of hypothesis imply a denser distribution of odd-order zeros of $ Z'(t) $, i.e., local extrema of $ \zeta(1/2 + it) $, than previously known?
• RQ2 Can the exponent in the interval length $ T^{\Delta} \psi(T) $ containing such zeros be improved beyond $ \Delta = 35/216 + \epsilon $, and if so, by how much?
• RQ3 To what extent does I.M. Vinogradov's scepticism about the method of trigonometric sums hold when applied to estimating the Hardy-Littlewood integral $ \int_0^T |\zeta(1/2 + it)| dt $?
• RQ4 Can a representation of the Hardy-Littlewood integral be constructed with an error term that is unconditionally negligible, independent of trigonometric sum estimates?

Key findings

• Under the Lindel"of hypothesis, the interval $ (T, T + T^{\epsilon} \psi(T)) $, where $ \psi(T) $ is any slowly diverging function (e.g., $ \ln\ln\cdots\ln T $), contains at least one odd-order zero of $ Z'(t) $, implying dense oscillation of $ Z(t) $.
• The exponent $ \Delta = 35/216 + \epsilon $ from prior work is improved to $ \epsilon $ under the Lindel"of hypothesis, representing a 100% improvement in the exponent bound.
• The number of odd-order zeros of $ Z'(t) $ in $ (T, T + T^{\tau}) $, for any fixed $ \tau > 0 $, grows at least as $ A(\tau,\epsilon) T^{\tau - \epsilon} $, confirming unbounded oscillation as $ T \to \infty $.
• The classical trigonometric sum method leads to an unbounded and unremovable error term $ Q(T) = \mathcal{O}(T^{1/4 + \epsilon}) $, confirming I.M. Vinogradov's scepticism about its long-term potential for finer estimates.
• A new asymptotic formula (A.6) for the Hardy-Littlewood integral is derived with error term $ \mathcal{O}(\ln T / T) $, which is negligible and independent of trigonometric sum estimates, confirming the validity of Vinogradov's scepticism in this context.
• The method of trigonometric sums has been shown to yield diminishing returns: improvements in exponent bounds (e.g., from $ 1/4 \to 1/6 \to 1/6 - \epsilon $) are increasingly small, suggesting a fundamental limit to its effectiveness.