[Paper Review] On the Riemann zeta-function and the divisor problem III

Aleksandar Ivić|ArXiv.org|Oct 18, 2006

Analytic Number Theory Research11 references4 citations

TL;DR

This paper investigates the error term $ R(T) $ in the integral of $ E^*(t) $, the refined error function linking the Riemann zeta-function and the Dirichlet divisor problem. By analyzing higher moments and applying advanced exponential sum techniques, the authors establish $ R(T) = O_{\varepsilon}(T^{593/912 + \varepsilon}) $, $ \int_0^T R^2(t)\,dt = T^2 P_3(\log T) + O_{\varepsilon}(T^{11/6 + \varepsilon}) $, and $ \int_0^T R^4(t)\,dt \ll_{\varepsilon} T^{3 + \varepsilon} $, where $ P_3 $ is a cubic polynomial with positive leading coefficient.

ABSTRACT

Let $Δ(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|ζ(1/2+it)|$. If $E^*(t) = E(t) - 2πΔ^*(t/2π)$ with $Δ^*(x) = -Δ(x) + 2Δ(2x) - {1\over2}Δ(4x)$ and we set $\int_0^T E^*(t) dt = 3πT/4 + R(T)$, then we obtain $$ R(T) = O_ε(T^{593/912+ε}), \int_0^TR^4(t) dt \ll_εT^{3+ε}, $$ and $$ \int_0^TR^2(t) dt = T^2P_3(\log T) + O_ε(T^{11/6+ε}), $$ where $P_3(y)$ is a cubic polynomial in $y$ with positive leading coefficient.

Motivation & Objective

• To refine the understanding of the error term $ R(T) $ arising from the integral of $ E^*(t) $, a hybrid function connecting the Riemann zeta-function and the Dirichlet divisor problem.
• To improve the trivial bound $ R(T) = O(T^{3/4}) $ by establishing a non-trivial, explicit error exponent.
• To determine the asymptotic behavior of the second and fourth moments of $ R(T) $, revealing polynomial-logarithmic structure in the main term.
• To investigate the conjectural size of $ R(T) $, with implications for the size of $ E^*(T) $ and $ \Delta^*(x) $, and to assess the limits of current analytic methods.

Proposed method

• Define $ R(T) $ via $ \int_0^T E^*(t)\,dt = \frac{3\pi}{4}T + R(T) $, where $ E^*(t) = E(t) - 2\pi\Delta^*(t/2\pi) $, with $ \Delta^* $ a smoothed version of the divisor problem error $ \Delta(x) $.
• Apply the method of moments, particularly leveraging bounds on $ \int_0^T |E^*(t)|^k\,dt $ for $ k=4,5 $ from prior work to derive moment estimates for $ R(T) $.
• Use exponential sum techniques and the saddle-point method to analyze the contribution of terms $ n \asymp T^{1/3} $ in the sum over $ d(n) $, focusing on the oscillatory behavior of $ \sqrt{n} $ terms.
• Employ Lemma 6 and Lemma 3 (with $ a=5 $) to control the error in integral approximations, particularly by truncating integration ranges and bounding $ \sin y / y $ terms.
• Replace finite integration intervals with $ (0, \infty) $, estimating the error via $ \sin^2 y / y^2 $ integrability and logarithmic moments.
• Establish the asymptotic form $ \int_0^T R^2(t)\,dt = T^2 P_3(\log T) + O_{\varepsilon}(T^{11/6 + \varepsilon}) $ by matching the structure of $ E^*(t) $'s second moment and extracting polynomial-logarithmic coefficients.

Experimental results

Research questions

• RQ1 What is the best possible error bound for $ R(T) = \int_0^T E^*(t)\,dt - \frac{3\pi}{4}T $, and how does it compare to the trivial $ O(T^{3/4}) $ bound?
• RQ2 Can the second moment $ \int_0^T R^2(t)\,dt $ be expressed as a main term involving a cubic polynomial in $ \log T $, and what is the error term?
• RQ3 What is the growth rate of the fourth moment $ \int_0^T R^4(t)\,dt $, and does it confirm or contradict the conjecture $ R(T) = O_{\varepsilon}(T^{1/2 + \varepsilon}) $?
• RQ4 How do the bounds on $ R(T) $ and its moments relate to the size of $ E^*(T) $, and what does this imply for the divisor problem and zeta-function mean square?
• RQ5 What is the role of the smoothed error function $ \Delta^*(x) $ in improving the analogy between the divisor problem and the zeta-function mean square?

Key findings

• The paper establishes $ R(T) = O_{\varepsilon}(T^{593/912 + \varepsilon}) $, where $ 593/912 \approx 0.6502 $, significantly improving the trivial $ O(T^{3/4}) $ bound.
• The second moment satisfies $ \int_0^T R^2(t)\,dt = T^2 P_3(\log T) + O_{\varepsilon}(T^{11/6 + \varepsilon}) $, where $ P_3(y) $ is a cubic polynomial in $ y $ with positive leading coefficient.
• The fourth moment satisfies $ \int_0^T R^4(t)\,dt \ll_{\varepsilon} T^{3 + \varepsilon} $, indicating that $ R(T) $ does not grow too rapidly in $ L^4 $-norm.
• The bound $ R(T) = \Omega(T^{1/2} (\log T)^{3/2}) $ is derived, suggesting that $ R(T) $ cannot be smaller than this order, supporting the conjecture $ R(T) = O_{\varepsilon}(T^{1/2 + \varepsilon}) $.
• The error exponent $ 593/912 $ arises from the limit of current analytic techniques, particularly the use of Lemma 2.5 and the saddle-point method in exponential sum estimation.
• The results imply that $ E^*(T) \ll_{\varepsilon} T^{1/4 + \varepsilon} $, which, if true, would refine the known asymptotic for the mean square of $ |\zeta(1/2 + it)| $.