AI 中文总结
该研究结合算术气体与熵-几何对应假设，导出量子修正黑洞度规，解释了黎曼假设的物理意义，得到带对数修正的黑洞热力学结果。
AI 中文摘要
黎曼ζ函数的欧拉乘积是自由玻色气体的配分函数，其模式能量为素数的对数。我们证明，该算术气体结合熵-几何对应假设，会导出量子修正黑洞度规。素数气体是Hagedorn系统，其熵与能量呈线性关系，恰好满足熵与视界面积线性相关的要求；ζ函数在β=1处的简单极点确定了面积定律对数修正的系数。非平凡零点无法发挥此作用：它们的能级密度仅对数增长，速度过慢，不满足广延性要求。要求重构几何在大半径处退化为史瓦西（Schwarzschild）度规，即可确定算术能量到视界面积的映射，得到闭式度规f(r)=1-2GMr/(r²+lz²)，其中lz²=αG/π。该度规描述具有Reissner-Nordström视界结构但无库仑毛发的双视界黑洞、满足零能量条件的正能量各向异性源、软化的中心奇点、有界的霍金温度，以及终止蒸发的冷极值残余。相同长度尺度可通过要求热力学第一定律与修正熵完全一致独立导出。非平凡零点仅作为面积中指数抑制的对数周期波纹存在，这提示黎曼假设的物理解释是算术对黑洞热力学的修正尽可能小。
英文摘要
The Euler product of the Riemann zeta function is the partition function of a free bosonic gas whose mode energies are the logarithms of the primes. We show that this arithmetic gas, combined with the assumption that the entropy--geometry correspondence holds, leads to a quantum-corrected black-hole metric. The prime gas is a Hagedorn system. Its entropy is linear in the energy, which is exactly what an entropy linear in the horizon area requires, and the simple pole of the zeta function at $β=1$ fixes the coefficient of the logarithmic correction to the area law. The nontrivial zeros cannot play this role: their level density grows only logarithmically, and far too slowly to be extensive. Demanding that the reconstructed geometry reduce to Schwarzschild at large radius then fixes the map from arithmetic energy to horizon area and yields the closed-form metric $f(r)=1-2GMr/(r^{2}+\lz^{2})$ with $\lz^{2}=αG/π$. This describes a two-horizon black hole with Reissner--Nordström horizon structure but no Coulombic hair, a positive-energy anisotropic source obeying the null energy condition, a softened central singularity, a bounded Hawking temperature, and a cold extremal remnant that ends the evaporation. The same length scale follows independently from requiring that the first law hold exactly with the corrected entropy. The nontrivial zeros survive only as exponentially suppressed log-periodic ripples in the area, which suggests a physical interpretation of the Riemann hypothesis as the statement that arithmetic corrections to black-hole thermodynamics are as small as they can be.
Comments9 pages, 2 figures
