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on Third-level log-concavity of the Riemann Xi kernel

The statement

Let

\Phi(r)=\sum_{m\ge1}\pi m^2\bigl(2\pi m^2e^{4r}-3\bigr)e^{5r-\pi m^2e^{4r}},\qquad r\ge0,

be the classical Riemann $\Xi$ -kernel, with its usual even extension, and put $S_0(t)=\Phi(\sqrt t)$ for $t>0$ . Recursively define

S_{j+1}(t)=(S_j'(t))^2-S_j(t)S_j''(t).

It is known that $S_0>0$ and $(\log S_0)''<0$ , and Planat--Solé (2026) proved $(\log S_1)''<0$ ; hence $S_1,S_2>0$ . Determine whether

S_3(t)>0\qquad\text{for every }t>0.

Equivalently, prove that $S_2$ is strictly log-concave on $(0,\infty)$ , i.e. $(\log S_2(t))''<0$ for all $t>0$ , or disprove this by giving a rigorous point or interval on which $S_3\le0$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.