bibkey: plewa2018laguerre
authors: Paweł Plewa
year: 2018
title: Sharp Hardy's type inequality for Laguerre expansions
doi: null
url: https://arxiv.org/abs/1810.08138v1
claim: The standard normalized Laguerre bulk estimate in equation (19) is uniform in degree and position; its parameter-one specialization controls the growing low-prime prefix of the actual centered Li row. The remaining middle interval and its signed joint estimate are not controlled.
strata_touched: []
license: citation-only
triage: anchor

A classical uniform Laguerre bound for the actual Li prefix

The inspected source is arXiv:1810.08138v1, 20 pages. Its PDF SHA-256 is 2251d296d1a523ba1873a990cf10c8d17f942f70792d80b6be0f8d5af42e59c6. The exact definitions and estimate were also read in the author’s TeX. Locators below refer to this manuscript. The journal metadata DOI 10.2969/jmsj/84448444 is not used to assert that an uninspected publisher edition has identical statements or numbering. The paper’s main Hardy inequality is not needed for this application.

Section 3’s convolution-function definition and Section 4’s standard-function definition give

$${\mathcal{L}}_k^{\alpha }(u)={\left(\frac{\Gamma (k+1)}{\Gamma (k+\alpha +1)}\right)}^{1/2}{u}^{\alpha /2}{e}^{-u/2}{L}_k^{(\alpha )}(u),\alpha >-1.$$

Section 4, printed p.12, equation (19), gives the classical bulk bound

$$|{\mathcal{L}}_k^{\alpha }(u)|\le C_{\alpha }(\nu u{)}^{-1/4},\nu =\max (4k+2\alpha +2,2),{\nu }^{-1}<u\le \nu /2.$$

For fixed $\alpha$, the constant is uniform in $k,u$. The manuscript attributes this estimate to Muckenhoupt, Mean convergence of Hermite and Laguerre series. II, printed p.435, and Askey–Wainger, Mean convergence of expansions in Laguerre and Hermite series, printed p.699. Those original proofs are not independently audited here. The estimate is directly reused, not reproved or counted as a new orthogonal-polynomial result.

At $k=n-1$ and $\alpha =1$, the exact normalization and parameter map are

$${\mathcal{L}}_{n-1}^1(u)=\sqrt{u/n}{e}^{-u/2}{L}_{n-1}^{(1)}(u),\nu =4n.$$

Hence the cited estimate supplies

$$|L_{n-1}^{(1)}(u)|\le C{n}^{1/4}{e}^{u/2}{u}^{-3/4},(4n{)}^{-1}<u\le 2n.$$

This range contains the entire growing interval $\log 2\le u\le \log N$, uniformly for integers $N/3\le n\le N$ and sufficiently large $N$. Thus the actual Li-prefix application uses a genuinely uniform bound, rather than a fixed-position asymptotic. It dominates the finite prime sum by an integer sum and obtains an $o(N)$ allowance for that specified prefix. No new prime-distribution estimate, cancellation, numerical validation, Lean certification or literature-priority claim is made. The remaining middle interval and its common-source signed comparison, Robin and RH remain unresolved.