A Fourth-Logarithmic-Derivative Inequality for the Riemann Xi Kernel

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Original abstract

We prove a global fourth-logarithmic-derivative inequality for the positive even Riemann Xi kernel \[\Phi(x)=\sum_{n=1}^{\infty}\pi n^2\left(2\pi n^2e^{4x}-3\right)\exp\left(5x-\pi n^2e^{4x}\right),\] extended evenly to the real axis. Writing \[L(x)=\log\Phi(x),\] the main result is \[\boxed{L''''(x)<0\qquad(x\in\mathbb R).}\] The proof is analytic and does not rely on interval arithmetic. After the substitution \[q=\pi e^{4x},\] the first three theta modes are isolated and reduced to an exact integer-polynomial inequality. The remaining infinite theta tail is controlled by explicit derivative majorants and elementary rational exponential bounds. Since \(L\) is even and \(L'''(0)=0\), the main theorem implies \[\boxed{L'''(x)<0\qquad(x>0).}\] This yields a further probabilistic/correlation consequence. Define \[Z(r)=\int_{\mathbb R}\Phi(y+r/2)\Phi(y-r/2)\,dy\] and the normalized overlap density \[p_r(y)=\frac{\Phi(y+r/2)\Phi(y-r/2)}{Z(r)}.\] Its variance, \[V(r)=\int_{\mathbb R}y^2p_r(y)\,dy,\] is proved to satisfy \[\boxed{V'(r)<0\qquad(r>0).}\] Thus the transverse width of the overlap of two shifted copies of the Riemann kernel strictly decreases with their separation. The associated second correlation kernel \[\nu_2(r)=\int_{\mathbb R}(r-2s)^2\Phi(r-s)\Phi(s)\,ds\] admits the exact factorization \[\boxed{\nu_2(r)=4Z(r)V(r).}\] Its Fourier transform is \[\boxed{\widehat{\nu_2}(t)=\frac1{32}\left[\Xi'(t/2)^2-\Xi(t/2)\Xi''(t/2)\right].}\] Using the correlation-kernel characterization of Dimitrov and Xu, the remaining off-axis condition in the present normalization is \[\boxed{\widehat{\cosh(ur)\nu_2(r)}(t)>0,\qquadt\in\mathbb R,\quad 0<|u|<1.}\] This condition is equivalent to the Riemann Hypothesis. Accordingly, the fourth-logarithmic-derivative inequality and the strict beat-variance monotonicity proved here are unconditional results, whereas extending them to the above global off-axis Fourier positivity would amount to proving RH. A companion paper establishes the separate finite-order total-positivity result \[\Phi\in\mathrm{PF}_4,\] indeed strict total positivity through order four: H. Umihara, *Strict Total Positivity of Order Four for the de Bruijn--Newman Kernel: An Exact Polynomial--Tail Proof of the Global PF4 Property*, Zenodo (2026), doi:10.5281/zenodo.21924561. That finite-order result is structurally related to the present work but does not imply the remaining RH-equivalent off-axis Fourier-positivity condition. The research was developed with substantial assistance from ChatGPT (OpenAI) for symbolic calculation, proof exploration, algebraic verification, and consistency checks. The human conceptual starting point was the “wave-integer” representation, in which an integer \(a\) is represented through the Fourier-dual pair \[\delta(x-a)\longleftrightarrowe^{iat},\] and the subsequent observation that \[e^{iat}+e^{ibt}=2e^{i(a+b)t/2}\cos\left(\frac{a-b}{2}t\right)\] may be analyzed as a beat phenomenon. This beat-wave viewpoint guided the transition from arithmetic wave representations to correlation kernels and the beat-variance structure studied in the present letter. The paper does not claim a proof of the Riemann Hypothesis.

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