[Paper Review] Jacob's ladders and the multiplicative asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral

Jan Moser|ArXiv.org|Jul 2, 2009

Analytic Number Theory Research4 references17 citations

TL;DR

This paper introduces a novel multiplicative asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral involving the square of the Hardy Z-function, $ Z^2(t) $, using Jacob's ladders—a geometric construction tied to the Riemann zeta function's zeros. The key result establishes that $ \int_T^{T+U} Z^2(t)\,dt = U\ln T \cdot \tan[\alpha(T,U)] \left\{1 + \mathcal{O}\left(\frac{\ln\ln T}{\ln T}\right)\right\} $ for $ U \leq T / \ln T $, offering a new class of asymptotic formulae unattainable by prior methods of Balasubramanian, Heath-Brown, and Ivić.

ABSTRACT

The elementary geometric properties of Jacob's ladders lead to a class of new asymptotic formulae for short and microscopic parts of the Hardy-Littlewood integral. This class of asymptotic formulae cannot be obtained by methods of Balasubramanian, Heath-Brown and Ivic.

Motivation & Objective

• To derive new asymptotic formulae for short and microscopic intervals of the Hardy-Littlewood integral $ \int_T^{T+U} Z^2(t)\,dt $, which are not accessible via classical methods.
• To establish a multiplicative asymptotic structure for $ Z^2(t) $ based on geometric properties of Jacob's ladders, linking analytic number theory with geometric constructions.
• To demonstrate that the asymptotic behavior of $ Z^2(t) $ on intervals $ [T, T+U] $ with $ U \to 0 $ can be captured via a multiplicative correction factor involving $ \ln T $ and the angle $ \alpha(T,U) $.
• To provide a new framework for analyzing the distribution of the zeta function's zeros by relating the integral of $ Z^2(t) $ to the derivative of the Jacob's ladder function $ \varphi(T) $.

Proposed method

• The method relies on the identity $ Z^2(t) = \Phi'_{\varphi}[\varphi(T)] \cdot \frac{d\varphi(T)}{dT} $, where $ \Phi'_{\varphi} $ is a transformed integral involving $ Z^2(t) $ and an exponential weight.
• The proof uses the definition of Jacob's ladder $ y = \frac{1}{2}\varphi(T) $, with $ \varphi(T) \sim 2T $, and the fundamental chord property $ \tan[\alpha(T,U_0)] = 1 + \mathcal{O}(1/\ln T) $.
• A key step involves estimating the second derivative $ \Phi''_{y^2}[\varphi] $, showing it is $ \mathcal{O}\left(\frac{1}{\varphi}\ln\varphi\ln\ln\varphi\right) $, which controls error terms.
• The asymptotic formula is derived by comparing the additive formula (1.1) with the new multiplicative structure, using the mean value theorem and error propagation through $ \Phi'_{\varphi} $.
• The method introduces a new class of intervals $ [N,M] \subset [T, T+U_0] $ where the average of $ Z^2(t) $ is asymptotically $ \ln T $, characterized geometrically by the 'almost parallel chord' property $ \tan[\alpha(N,M-N)] = 1 + o(1) $.
• The framework is extended to intervals starting at individual non-trivial zeros $ \gamma $ of $ \zeta(1/2 + it) $, yielding $ \int_{\gamma}^{\gamma+U} Z^2(t)\,dt \sim U\ln\gamma \cdot \tan\alpha $ for $ \tan\alpha \in [\eta, 1-\eta] $.

Experimental results

Research questions

• RQ1 Can a multiplicative asymptotic formula be derived for the short and microscopic parts of the Hardy-Littlewood integral $ \int_T^{T+U} Z^2(t)\,dt $ when $ U \to 0 $?
• RQ2 Does the geometric structure of Jacob's ladders—specifically the angle of the chord—determine the asymptotic behavior of $ Z^2(t) $ on small intervals?
• RQ3 Can the asymptotic formula for $ \int_T^{T+U} Z^2(t)\,dt $ be expressed in a multiplicative form involving $ \ln T $ and $ \tan[\alpha(T,U)] $, independent of the additive methods of Balasubramanian, Heath-Brown, and Ivić?
• RQ4 Is there a continuum of intervals $ [N,M] $ with $ M-N < 1 $ for which $ \int_N^M Z^2(t)\,dt \sim (M-N)\ln T $?
• RQ5 What is the size of the error term in the multiplicative asymptotic formula, and can it be bounded uniformly across all $ U \leq T / \ln T $?

Key findings

• The main result is the multiplicative asymptotic formula: $ \int_T^{T+U} Z^2(t)\,dt = U\ln T \cdot \tan[\alpha(T,U)] \left\{1 + \mathcal{O}\left(\frac{1}{\ln T}\right)\right\} $ for $ U \in \left(0, \frac{T}{\ln T}\right] $.
• The formula implies that $ Z^2(T) = \frac{1}{2}\ln T \cdot \frac{d\varphi(T)}{dT} \left\{1 + \mathcal{O}\left(\frac{\ln\ln T}{\ln T}\right)\right\} $, providing a pointwise asymptotic for $ Z^2(T) $.
• There exists a continuum of intervals $ [N,M] \subset [T, T+U_0] $ such that $ \int_N^M Z^2(t)\,dt = (M-N)\ln T \left\{1 + \mathcal{O}\left(\frac{\ln\ln T}{\ln T}\right)\right\} $, including intervals with $ M-N < 1 $.
• For every sufficiently large zero $ \gamma $ of $ \zeta(1/2 + it) $, there is a continuum of intervals $ [\gamma, \gamma + U] $ such that $ \int_\gamma^{\gamma+U} Z^2(t)\,dt = U\ln\gamma \cdot \tan\alpha \left\{1 + \mathcal{O}\left(\frac{1}{\ln \gamma}\right)\right\} $ for $ \tan\alpha \in [\eta, 1-\eta] $.
• The second derivative $ \Phi''_{y^2}[\varphi] $ is bounded by $ \mathcal{O}\left(\frac{1}{\varphi}\ln\varphi\ln\ln\varphi\right) $, which ensures the error terms remain controlled in the asymptotic expansion.
• The method yields results that cannot be derived using the complex analytic techniques of Balasubramanian, Heath-Brown, and Ivić, particularly for microscopic intervals where $ U \to 0 $.