[Paper Review] On the integral of the error term in the Dirichlet divisor problem

Aleksandar Ivić|ArXiv.org|Oct 6, 2005

Analytic Number Theory Research9 references3 citations

TL;DR

This paper investigates the integral of the error term Δₖ(x) in the generalized Dirichlet divisor problem using Perron's formula and Mellin transforms. It establishes sharp bounds for ∫₁ˣ Δₖ(u) du and its mean square, particularly proving that ∫₁ˣ Δ₂(u) du ≪ x^{3/4} and ∫₁ˣ (∫₁ᵘ Δ₂(t) dt)² du ∼ Cx^{5/2}, confirming the conjectured growth rate for the classical divisor problem.

ABSTRACT

Several results are obtained concerning the function $Δ_k(x)$, which represents the error term in the general Dirichlet divisor problem. These include the estimates for the integral of this function, as well as for the corresponding mean square integral. The mean square integral of $Δ_2(x)$ is investigated in detail.

Motivation & Objective

• To analyze the integral of the error term Δₖ(x) in the generalized Dirichlet divisor problem for small k.
• To derive pointwise and mean square estimates for ∫₁ˣ Δₖ(u) du using complex analysis techniques.
• To investigate the behavior of the mean square integral of Δ₂(x), the classical divisor problem error term.
• To explore the possibility of sharper asymptotic expansions for Δ₂(x) beyond known bounds.
• To examine the implications of mean square results for conjectures on the size of Δ(x) and its connection to the Lindelöf hypothesis.

Proposed method

• Applies the Perron inversion formula to express Δₖ(x) as a complex integral involving ζᵏ(s) and xˢ/s.
• Integrates the Perron formula to derive an expression for ∫₁ˣ Δₖ(u) du in terms of a Mellin-type integral involving ζᵏ(s)/(s(s+1)).
• Uses power moment estimates for ζ(s) to bound the integral of Δₖ(u) du, particularly for k=2,3.
• Employs the complex integration method and the first derivative test to estimate oscillatory integrals arising from Voronoi-type explicit formulas.
• Applies the Cauchy-Schwarz inequality and known L² bounds for the error function F(x) to control the mean square of ∫Δ₂(u) du.
• Uses analytic continuation and integral estimates to derive bounds for the Mellin transform of Δ₂(x) in the critical strip.

Experimental results

Research questions

• RQ1 What is the best possible upper bound for ∫₁ˣ Δₖ(u) du for small k, particularly k=2 and k=3?
• RQ2 How does the mean square integral of ∫₁ˣ Δ₂(u) du grow as x → ∞?
• RQ3 Can the explicit formula for Δ₂(x) via Bessel functions be used to derive precise asymptotic estimates for its integral?
• RQ4 What is the optimal exponent α such that ∫₁ˣ Δ₂(u) du = O(x^{α})?
• RQ5 Is it possible to improve the conjectural bound Δ(x) ≪ x^{1/4+ε} using mean square information?

Key findings

• For k=2, the integral ∫₁ˣ Δ(u) du satisfies ∫₁ˣ Δ(u) du ≪ x^{3/4}, and the mean square integral ∫₁ˣ (∫₁ᵘ Δ(t) dt)² du ∼ Cx^{5/2} as x → ∞, with C > 0.
• For k=3, the integral ∫₁ˣ Δ₃(u) du ≪ε x^{1+ε}, and the corresponding mean square integral is ≪ε x^{3+ε}.
• The mean square of the integral of Δ₂(u) du is shown to be ≪ T^{6-4σ}(log T)^{12-8σ} for 1 < σ < 3/2, confirming convergence in the critical strip.
• The paper proves that if Δ(x) ≪ x^{1/4+ε} holds, then the conjectural bound α > 3/4 in F(x) = -1/(4π²)x log²x + κx log x + λx + G(x), G(x) = O(x^α), is optimal.
• The bound α > 3/4 is shown to be necessary, as assuming α < 3/4 leads to a contradiction via the Cauchy-Schwarz inequality and the known lim sup |Δ(x)x^{-1/4}| = ∞.
• The results support the conjecture that Δ(x) ≪ x^{1/4+ε} is sharp, and that a refined asymptotic expansion for F(x) with α = 3/4 + ε is plausible.