[Paper Review] On some mean square estimates for the zeta-function in short intervals

Aleksandar Ivić|arXiv (Cornell University)|Dec 4, 2012

Analytic Number Theory Research8 references3 citations

TL;DR

This paper establishes improved lower bounds for the mean square of the error term $ E^*(t) $ and its integral $ R(t) $ in short intervals of the Riemann zeta-function, using exponential sum estimates and Voronoï-type formulas. It proves that $ \int_T^{T+H} (E^*(t))^2 dt \gg HT^{1/3}\log^3 T $ and $ \int_T^{T+H} R^2(t) dt \gg HT\log^3 T $ for $ H \geq T^{2/3+\varepsilon} $, extending previous ranges and confirming the persistence of large values of these error functions in short intervals.

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 $Δ^*(x) = -Δ(x) + 2Δ(2x) - 1/2Δ(4x)$ and we set $\int_0^T E^*(t)\,dt = 3πT/4 + R(T)$, then we obtain $$ \int_T^{T+H}(E^*(t))^2\,dt \gg HT^{1/3}\log^3T $$ and $$ HT\log^3T \ll \int_T^{T+H}R^2(t)\,dt \ll HT\log^3T, $$ for $T^{2/3+ε}\le H \le T$.

Motivation & Objective

• To improve the range of $ H $ for which lower bounds on the mean square of $ E^*(t) $ and $ R(t) $ hold in short intervals $[T, T+H]$.
• To establish that $ E^*(t) $ and $ R(t) $ exhibit large oscillations in short intervals, confirming they cannot be too small.
• To extend the validity of asymptotic formulas for $ \int_0^T (E^*)^2 dt $ and $ \int_0^T R^2(t) dt $ to shorter intervals via new estimates.
• To investigate the size distribution of $ E^*(t) $ and $ R(t) $, particularly the existence of large positive and negative values in short intervals.

Proposed method

• Use of Atkinson’s explicit formula for $ E(T) $ and Voronoï-type formulas for $ \Delta^*(x) $ to express $ E^*(t) $ and $ R(t) $ in terms of oscillatory exponential sums.
• Application of two-dimensional exponential sum estimates to control error terms in the asymptotic expansions of $ E^*(t) $ and $ R(t) $.
• Estimation of oscillatory integrals involving $ \sin^2(\cdots) $ and $ \sin(\cdots) $ terms using trigonometric identities and bounds on $ \sin x $.
• Decomposition of sums into dyadic intervals $ K < n \leq 2K $ and estimation of $ L^2 $-norms of exponential sums via $ \ll_{\varepsilon} T^{1/2+\varepsilon}(HK^{3/2} + T^{1/2}K^2) $ and similar bounds.
• Use of the identity $ 1 - \cos \gamma = 2\sin^2(\gamma/2) $ and $ |\sin x| \geq \frac{2}{\pi}|x| $ for $ |x| \leq \pi/2 $ to extract main terms from oscillatory contributions.
• Comparison of the main term in $ E^*(t) $ with $ \sum_{n \ll T^{1/3}} d^2(n) n^{-3/2} \int_T^{T+H} t^{1/2} \sin^2(\cdots) dt $, leading to a lower bound of order $ HT^{1/3}\log^3 T $.

Experimental results

Research questions

• RQ1 Can the range of $ H $ for which $ \int_T^{T+H} (E^*(t))^2 dt \gg HT^{1/3}\log^3 T $ holds be improved beyond $ H \geq T^{5/6+\varepsilon} $?
• RQ2 Do the error terms $ E^*(t) $ and $ R(t) $ exhibit large positive and negative values in short intervals of length $ H \geq T^{2/3+\varepsilon} $?
• RQ3 Is the error term in the asymptotic formula for $ \int_0^T R^2(t) dt $ improvable beyond $ O_{\varepsilon}(T^{11/6+\varepsilon}) $?
• RQ4 Can the lower bounds for $ \int_T^{T+H} R^2(t) dt $ be established for $ H \geq T^{2/3+\varepsilon} $, matching the improved range for $ E^*(t) $?
• RQ5 What is the precise size of $ E^*(t) $ and $ R(t) $ in short intervals, and do they satisfy $ \Omega $-results similar to those for $ E(t) $ and $ \Delta(x) $?

Key findings

• For $ T^{2/3+\varepsilon} \leq H \leq T $, the mean square of $ E^*(t) $ satisfies $ \int_T^{T+H} (E^*(t))^2 dt \gg HT^{1/3}\log^3 T $, improving the range of previous lower bounds.
• For the same range of $ H $, the mean square of $ R(t) $ satisfies $ \int_T^{T+H} R^2(t) dt \gg HT\log^3 T $, extending the validity of the $ \Omega $-result for $ R(t) $.
• The upper bound $ \int_T^{T+H} R^2(t) dt \ll_{\varepsilon} HT\log^3 T + T^{5/3+\varepsilon} $ holds for $ T^{\varepsilon} \leq H \leq T $, confirming the growth rate in short intervals.
• The result implies that every interval $[T, T+H]$ with $ H = T^{2/3+\varepsilon} $ contains points where $ |E^*(t)| \gg t^{1/6}\log^{3/2}t $ and $ |R(t)| \gg t^{1/2}\log^{3/2}t $, confirming non-vanishing oscillations.
• The proof relies on estimating exponential sums and extracting main terms from oscillatory integrals, with error terms controlled via $ O_{\varepsilon}(T^{1+\varepsilon}) $ bounds.
• The asymptotic structure of $ R(t) $ is shown to be governed by a sum over $ n \leq T^{1/2-\varepsilon} $, with the dominant contribution arising from $ n \ll T^{1/3} $, leading to the lower bound.