AI 中文总结
作者通过两种互补方法证明了Csordas等人提出的关于黎曼Ξ核二阶凹性的猜想，该结果蕴含相关双重Turán不等式但不涉及黎曼假设。
AI 中文摘要
设Φ为黎曼Ξ函数傅里叶表示中的经典雅可比θ核，令s(t)=Φ(√t)，定义第一个拉盖尔表达式f(t)=s’(t)²−s(t)s''(t)。Csordas和Dimitrov于2000年猜想log f在(0,∞)上严格凹；Csordas于2015年将该断言重述为开放问题4.14。我们通过两种互补方法证明该猜想，两种方法仅在模不动点附近共享一个局部证书。第一种证明是直接的θ级数论证：将t=0附近的定向舍入泰勒证书与带有严格θ尾界的首项主导估计相结合。第二种证明利用θ₃的雅可比非线性三阶微分方程，得到三维自治相空间、严格椭圆单调性定理以及目标不等式的精确四次约化。该四次边界是二次锥XY=Z²的多项式剪切。定向区间证书证明，在唯一剩余的紧区间上的每一次可能的锥接触都严格指向期望区域；超出该区间后，逐点单调性定理完成论证。有限证书和精确符号检查以可复现脚本形式提供。该结果通过Csordas-Dimitrov定理蕴含相关的双重Turán不等式，但未对黎曼假设作出任何断言。
英文摘要
Let $Φ$ be the classical Jacobi-theta kernel in the Fourier representation of the Riemann $Ξ$-function, set $s(t)=Φ(\sqrt t)$, and define the first Laguerre expression $f(t)=s'(t)^2-s(t)s''(t)$. Csordas and Dimitrov (2000) conjectured that $\log f$ is strictly concave on $(0,\infty)$; Csordas (2015) later restated the assertion as Open Problem~4.14. We prove the conjecture by two complementary methods, sharing only a local certificate near the modular fixed point. The first proof is a direct theta-series argument: a directed-rounding Taylor certificate near $t=0$ is joined to a dominant-first-summand estimate with rigorous theta-tail bounds. The second proof uses Jacobi's nonlinear third-order differential equation for $θ_3$ to obtain a three-dimensional autonomous phase space, a sharp elliptic monotonicity theorem, and an exact quartic reduction of the target inequality. The quartic boundary is a polynomial shear of the quadratic cone $XY=Z^2$. A directed interval certificate proves that every possible cone contact on the only remaining compact interval points strictly into the desired region; beyond that interval a pointwise monotonicity theorem closes the argument. The finite certificates and exact symbolic checks are supplied as reproducible scripts. The result implies the associated double Turán inequalities through the theorem of Csordas--Dimitrov, but no assertion of the Riemann Hypothesis is made.
Comments18 pages, 2 figures
