The long‐standing conjecture that for p∈(1,∞)$p \in (1, \infty)$ the ℓp(Z)$\ell ^p(\mathbb {Z})$ norm of the Riesz–Titchmarsh discrete Hilbert transform is the same as the Lp(R)$L^p(\mathbb {R})$ norm of the classical Hilbert transform, is verified when p=2n$p = 2 n$ or pp−1=2n$\frac{p}{p - 1} = 2 n$, for n∈N$n \in \mathbb {N}$. The proof, which is algebraic in nature, depends in a crucial way on the sharp estimate for the ℓp(Z)$\ell ^p(\mathbb {Z})$ norm of a different variant of this operator for the full range of p$p$. The latter result was recently proved by the authors (Duke Math. J. 168 (2019), no. 3, 471–504). [ABSTRACT FROM AUTHOR]