Asymptotics of Landau Constants with Optimal Error Bounds

Abstract

We study the asymptotic expansion for the Landau constants $G_n$,

$$\begin{aligned} \pi G_n\sim \ln N + \gamma +4\ln 2 + \sum _{s=1}^\infty \frac{\beta _{2s}}{ N^{2s}},\quad n\rightarrow \infty , \end{aligned}$$

where $N=n+3/4, \gamma =0.5772\ldots$ is Euler’s constant, and $(-1)^{s+1}\beta _{2s}$ are positive rational numbers, given explicitly in an iterative manner. We show that the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative $n$. Consequently, we obtain optimal sharp bounds up to arbitrary orders of the form

$$\begin{aligned} \ln N+\gamma +4\ln 2+\sum _{s=1}^{2m}\frac{\beta _{2s}}{N^{2s}}< \pi G_n < \ln N+\gamma +4\ln 2+\sum _{s=1}^{2k-1}\frac{\beta _{2s}}{N^{2s}} \end{aligned}$$

for all $n=0,1,2,\ldots , m=1,2,\ldots$, and $k=1,2,\ldots$. The results are proved by approximating the coefficients $\beta _{2s}$ with the Gauss hypergeometric functions involved and by using the second-order difference equation satisfied by $G_n$, as well as an integral representation of the constants $\rho _k=(-1)^{k+1}\beta _{2k}/(2k-1)!$.