Let Phi be the classical Jacobi-theta kernel in the Fourier representation of the Riemann Xi-function. Define s(t)=Phi(√t) and f(t)=s'(t)^2-s(t)s''(t) for t>0. The conjecture asserts that f(t)>0 and that log f is strictly concave on (0,∞); equivalently, fracd^2dt^2log f(t) 0.

Csordas–Dimitrov second-level concavity conjecture

Let $\Phi$ be the classical Jacobi-theta kernel in the Fourier representation of the Riemann $\Xi$ -function. Define $s(t)=\Phi(\sqrt{t})$ and $f(t)=s'(t)^2-s(t)s''(t)$ for $t>0$ . The conjecture asserts that $f(t)>0$ and that $\log f$ is strictly concave on $(0,\infty)$ ; equivalently, $\frac{d^2}{dt^2}\log f(t)<0$ for every $t>0$ .

Progress summary

Solved

An August 2026 unrefereed preprint claims to prove the conjecture, but no independent verification was found.

The conjecture asserts a second-level concavity property for the Riemann Xi kernel. It would imply related double Turán inequalities, but not the Riemann hypothesis.

August 2026 claimed proof

An arXiv manuscript claims two analytic proofs, supported by directed-rounding interval certificates and exact symbolic checks. This is a claim in an unrefereed preprint; no independent verification or objection is recorded in the retrieved source.

Current status (as of August 2026): The conjecture has a claimed but unverified proof in an unrefereed preprint; its correctness remains unsettled.

Sources