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

Abstract

We study the asymptotic expansion for the Landau constants $G_n$, \begin{equation*} πG_{n}\sim \ln(16N)+γ+\sum^{\infty}_{k=1}\frac{α_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 $(-1{)}^{\frac{l(l+1)}{2}}{\epsilon }_l(N)<0$ for $l=0,1,2,\cdots$ and $n=0,1,2,\cdots$, ${\epsilon }_l(N)$ 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)+γ+\sum_{k=1}^{p}\frac{α_{k}}{N^{k}}<πG_{n}<\ln(16N)+γ+\sum_{k=1}^{q}\frac{α_{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 $m=0,1,2,\cdots$.

17 pages, 1 figure. Proof of Theorem 2 simplified as compared with V1

References in corpus (1)

• Asymptotics of Landau constants with optimal error bounds