Proof of a conjecture of Granath on optimal bounds of the Landau constants

TL;DR

This paper proves a conjecture by Granath regarding the signs of coefficients in the asymptotic expansion of Landau constants and derives sharp bounds for these constants using the proven properties.

Contribution

It establishes the sign pattern of the asymptotic expansion coefficients and confirms Granath's conjecture, enabling precise bounds for Landau constants.

Findings

Confirmed the sign pattern of the asymptotic expansion coefficients.

Proved Granath's conjecture on the error terms.

Derived sharp bounds for Landau constants at arbitrary orders.

Abstract

We study the asymptotic expansion for the Landau constants $G_n$, \begin{equation*} \pi G_{n}\sim \ln(16N)+\gamma+\sum^{\infty}_{k=1}\frac{\alpha_k}{N^k} ~~\mbox{as} ~ n\rightarrow\infty, \end{equation*} where $N=n+1$, and $\gamma$ is Euler's constant. We show that the signs of the coefficients $\alpha_{k}$ demonstrate a periodic behavior such that $(-1)^{\frac {l(l+1)} 2} \alpha_{l+1}< 0$ for all $l$. We further prove a conjecture of Granath which states that $(-1)^{\frac {l(l+1)} 2} \varepsilon_l(N)<0$ for $l=0,1,2,\cdots$ and $n=0,1,2,\cdots$, $\varepsilon_l(N)$ being the error due to truncation at the $l$-th order term. Consequently, we also obtain the sharp bounds up to arbitrary orders of the form \begin{equation*} \ln(16N)+\gamma+\sum_{k=1}^{p}\frac{\alpha_{k}}{N^{k}}<\pi G_{n}<\ln(16N)+\gamma+\sum_{k=1}^{q}\frac{\alpha_{k}}{N^{k}} \end{equation*} for all $n=0,1,2\cdots$, all $p=4s+1,\; 4s+2$ and $q=4m,\; 4m+3$, with $s=0,1,2,\cdots$ and $m=0, 1, 2,\cdots$.