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Let $T_{N}$ be the (Hilbert) matrix defined by $T_{N}(m,n)=\frac{1}{m-n}$ if $1\leq m,n \leq N$ and $m\neq n$ , and $ T_{N}(n,n)=0$ if $1\leq n \leq N$ . It's well known that $\Vert T_{N}\Vert < \pi $. In a work with O.Leveque we found that there are positive constants $a$ and $b$ such that $\frac{a}{N} < \pi - \Vert T_{N}\Vert < \frac{b \log N}{N}$ , ($N>1$) . Does somebody know the exact order for $ \pi - \Vert T_{N}\Vert$ ?

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