[Paper Review] A generalization of Gallagher's lemma for exponential sums

Giovanni Coppola, Maurizio Laporta|arXiv (Cornell University)|Oct 21, 2014

Analytic Number Theory Research9 references3 citations

TL;DR

This paper generalizes Gallagher's lemma for exponential sums by introducing a weight function into the mean square estimate, enabling improved bounds for Dirichlet polynomials and Selberg integrals. The key contribution is a weighted inequality that enhances smoothing via self-convolutions, with the Cesàro weight yielding tighter estimates than the box or Lanczos weights under certain conditions.

ABSTRACT

First we generalize a famous lemma of Gallagher on the mean square estimate for exponential sums by plugging a weight in the right hand side of Gallagher's original inequality. Then we apply it in the special case of the Cesaro weight, in order to establish some results mainly concerning the classical Dirichlet polynomials and the Selberg integrals of an arithmetic function $f$, that are tools for studying the distribution of $f$ in short intervals. Furthermore, we describe the smoothing process via self-convolutions of a weight, that is involved into our Gallagher type inequalities, and compare it with the analogous process via the so-called correlations. Finally, we discuss a comparison argument in view of refinements on the Gallagher weighted inequalities according to different instances of the weight.

Motivation & Objective

• To extend Gallagher’s classical mean square estimate for exponential sums by incorporating a general weight function in the right-hand side of the inequality.
• To analyze the implications of this generalization for Dirichlet polynomials and Selberg integrals of arithmetic functions, particularly in short interval distribution studies.
• To compare the effectiveness of different smoothing weights—Cesàro, Lanczos, and box—via their Fourier transforms and minimal spectral norms over intervals.
• To establish conditions under which the weighted inequality yields sharper bounds than the original Gallagher lemma, especially in the context of moment estimates for the Riemann zeta function.
• To lay the foundation for future extensions to weighted Selberg integrals $ J_{w,f}(N,H) $ under general weight conditions.

Proposed method

• Derives a generalized inequality (⋆⋆) by inserting a weight $ w o w_ ho $ into the mean square estimate, with the key term $ m_{ ho,T} = ext{ess} ext{inf}_{|t| eq T} | ilde{w}_ ho(t)|^2 $, ensuring a lower bound on the Fourier transform magnitude.
• Applies the inequality to the Cesàro weight $ C_ ho(y) = ext{max}(1 - ho^{-1}|y|, 0) $, yielding a new estimate (⋆~) that improves upon the original Gallagher bound via averaging over short intervals.
• Compares the smoothing effect of self-convolution $ w * w $ with that of correlations, showing that self-convolution leads to better control of the $ L^2 $-norm of exponential sums.
• Uses the Hardy-Littlewood majorant principle to relate bounds on $ |s( u)| $ to bounds on $ b( u) $, enabling comparison of weighted sums via non-negative self-convolutions.
• Analyzes the Lanczos weight $ ilde{L}_{ ho, ho} $ and compares its Fourier transform magnitude with that of the box and Cesàro weights, showing asymptotic improvement as $ ho T o 1/2 $.
• Demonstrates that for $ 0 < ho T < 1/2 $, the Lanczos weight $ ilde{L}_{ ho, ho} $ is almost $ T $-better than the box weight $ extbf{1}_ ho $, and the Cesàro weight $ C_ ho $ is almost $ T $-better than $ ilde{L}_{ ho, ho} $.

Experimental results

Research questions

• RQ1 How can Gallagher’s classical mean square estimate for exponential sums be generalized to include an arbitrary weight function?
• RQ2 What are the implications of this generalization for the study of Dirichlet polynomials and Selberg integrals of arithmetic functions?
• RQ3 How do different smoothing weights—Cesàro, Lanczos, and box—compare in terms of their spectral norms and resulting bounds on exponential sums?
• RQ4 Under what conditions does the Cesàro weight yield a strictly better bound than the box or Lanczos weights in the context of $ L^2 $-norm estimates?
• RQ5 Can the generalized inequality be extended to weighted Selberg integrals $ J_{w,f}(N,H) $, and what are the implications for moment estimates of the Riemann zeta function?

Key findings

• The generalized inequality (⋆⋆) holds for any locally integrable weight $ w o w_ ho $, with the constant $ m_{ ho,T} = ext{ess} ext{inf}_{|t| eq T} | ilde{w}_ ho(t)|^2 $, providing a lower bound on the $ L^2 $-norm of exponential sums.
• For the Cesàro weight $ C_ ho $, the inequality yields $ igracevert S igracevert_{2,T}^2 riangleq igracevert ext{sum } s( u)e( u t) igracevert_{L^2(-T,T)}^2 riangleq ext{ess} ext{inf}_{|t| eq T} | ilde{C}_ ho(t)|^2 imes ext{integral of } | ext{sum } s( u)C_ ho( u - x)|^2 dx $, improving upon the original Gallagher bound.
• The Cesàro weight $ C_ ho $ is shown to be almost $ T $-better than the Lanczos weight $ ilde{L}_{ ho, ho} $ under the condition $ 0 < ho T < 1/2 $, due to a more favorable ratio of Fourier transform magnitudes.
• The Lanczos weight $ ilde{L}_{ ho, ho} $ is almost $ T $-better than the box weight $ extbf{1}_ ho $ under the same condition, indicating a hierarchy of smoothing efficiency.
• The method enables a new approach to bounding the Selberg integral $ J_f(N,h) $: assuming $ ilde{J}_f(N,h) riangleq ext{modified Selberg integral} riangleq ext{bound} ext{ for } f $, then $ J_f(N,h) riangleq ext{unconditional lower bound} $, with a concrete example: $ J_3(N,h) riangleq ext{Selberg integral for } d_3 riangleq ext{divisor function} $, and under the hypothesis $ ilde{J}_3(N,h) riangleq ext{bound} riangleq N^{1+ ho}h $, one obtains $ J_3(N,h) riangleq ext{bound} riangleq N^{1+ ho}h^{6/5} $.
• The improved bounds suggest a potential path to the weak sixth moment of the Riemann zeta function, assuming the conjectural bound $ ilde{J}_3(N,h) riangleq N^{1+ ho}h $ for $ h riangleq ext{small} $, which would imply a non-trivial upper bound on the sixth moment.