  The long-standing conjecture that for $p \in (1, \infty)$ the $\ell^p(\mathbb
Z)$ norm of the Riesz--Titchmarsh discrete Hilbert transform is the same as the
$L^p(\mathbb R)$ norm of the classical Hilbert transform, is verified when $p =
2 n$ or $\frac{p}{p - 1} = 2 n$, for $n \in \mathbb N$. The proof, which is
algebraic in nature, depends in a crucial way on the sharp estimate for the
$\ell^p(\mathbb Z)$ norm of a different variant of this operator for the full
range of $p$. The latter result was recently proved by the authors in
[Ba\~nuelos, Kwa\'snicki, On the $\ell^p$-norm of the discrete Hilbert
transform, Duke Math. J. 168(3) (2019): 471-504].
