[Paper Review] Jacob's ladders and the asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral of the function $|\zeta(1/2+it)|^4$

Jan Moser|arXiv (Cornell University)|Jan 22, 2010

Analytic Number Theory Research12 references10 citations

TL;DR

This paper introduces a novel asymptotic formula for short and microscopic segments of the Hardy-Littlewood integral of |ζ(1/2 + it)|⁴ using second-order Jacob's ladders, a geometric construction derived from the zeta function's behavior. By analyzing the angle of chords on the Jacob's ladder curve ϕ₂(T), the authors derive new asymptotic expressions that surpass existing methods by Balasubramanian, Heath-Brown, and Ivić, particularly for intervals of length U ≤ T¹³/¹⁴⁺²ǫ.

ABSTRACT

The elementary geometric properties of Jacob's ladders of the second order lead to a class of new asymptotic formulae for short and microscopic parts of the Hardy-Littlewood integral of $|\zeta(1/2+it)|^4$. These formulae cannot be obtained by methods of Balasubramanian, Heath-Brown and Ivic.

Motivation & Objective

• To derive new asymptotic formulae for short and microscopic segments of the Hardy-Littlewood integral ∫|ζ(1/2+it)|⁴dt.
• To overcome limitations of existing methods (e.g., Balasubramanian, Heath-Brown, Ivić) in analyzing small-scale behavior of the zeta function's fourth moment.
• To establish a geometric link between the asymptotic behavior of the zeta function and the curvature properties of Jacob's ladders of the second order.
• To provide a new framework for understanding the distribution of values of |ζ(1/2+it)|⁴ in short intervals and near zeros of the zeta function.

Proposed method

• Utilizes second-order Jacob's ladders ϕ₂(T), defined as the inverse of the cumulative distribution of the zeta function's fourth moment.
• Applies the identity Z⁴(t) = (1/(2π²)) · (1 + O((ln ln T)² / ln T)) · ln⁴T · dϕ₂(t)/dt on intervals [T, T+U₀] with U₀ = T¹³/⁴⁺²ǫ.
• Introduces the angle α₂(T,U) of the chord connecting (T, ϕ₂(T)) and (T+U, ϕ₂(T+U)) on the ϕ₂(T) curve to parameterize asymptotic behavior.
• Defines 'almost parallel chords' via tan[α₂(N,M−N)] = 1 + o(1), which correspond to intervals where the average of |ζ|⁴ is asymptotically constant.
• Analyzes microscopic regions near zeros γ of ζ(1/2+iT) by studying the inflection point ρ of ϕ₂(T) and the angle β(γ,ρ) of the chord from (γ,ϕ₂(γ)) to (ρ,ϕ₂(ρ)).
• Derives asymptotic formulae using tan(α) and tan(β) as scaling factors in the integral expressions, valid for U < ρ−γ or U < ḡ−γ.

Experimental results

Research questions

• RQ1 Can asymptotic formulae for short and microscopic parts of ∫|ζ(1/2+it)|⁴dt be derived beyond the reach of classical analytic methods?
• RQ2 How do geometric properties of second-order Jacob's ladders ϕ₂(T) relate to the distribution of |ζ(1/2+it)|⁴ in small intervals?
• RQ3 What role does the angle of the chord on ϕ₂(T) play in determining the asymptotic average of |ζ|⁴ over intervals of length U?
• RQ4 Can the behavior of |ζ|⁴ near zeros of ζ(1/2+iT) be captured via geometric features of ϕ₂(T)?
• RQ5 Are there continuum families of intervals where the average of |ζ|⁴ is asymptotically proportional to ln⁴T?

Key findings

• The asymptotic formula for short intervals is ∫ₜ⁺ᵘₜ |ζ(1/2+it)|⁴ dt = (1/(2π²)) · (1 + O((ln ln T)² / ln T)) · U ln⁴T · tan[α₂(T,U)], valid for U ≤ T¹³/¹⁴⁺²ǫ.
• The average value of |ζ(1/2+it)|⁴ over intervals [M,N] ⊂ [T,T+U₀] is asymptotically (1/(2π²)) ln⁴T if the chord on ϕ₂(T) is almost parallel to the fundamental chord.
• There exists a continuum of intervals [M,N] with M−N < 1 where ∫ₘₙ |ζ|⁴ dt ∼ (1/(2π²)) (M−N) ln⁴T, implying local equidistribution of the zeta function's fourth moment.
• Near a zero γ of ζ(1/2+iT), microscopic intervals [γ,γ+U] support asymptotic formulae ∫ₜ⁺ᵘₜ |ζ|⁴ dt ∼ (tan α)/(2π²) U ln⁴γ, with tan α ∈ (0, β(γ,ρ)) and U < ρ−γ.
• For intervals near γ with length U ≤ ḡ−γ, where ḡ = γ + γ¹³/¹⁴⁺²ǫ + ∆(γ), the formula ∫ₘₙ |ζ|⁴ dt ∼ (1/(2π²)) (M−N) ln⁴γ holds for chords parallel to the fundamental chord.
• All derived formulae, including (4.4) and (4.5), are provably unattainable via Balasubramanian, Heath-Brown, and Ivić's methods, as explicitly stated in Remark 7.