AI 中文总结
本文证明Riemann zeta函数零点及有限域簇zeta函数对数导数的Cauchy极限定律，给出显式误差界，并建立平稳点测度Stieltjes变换的收敛定理。
AI 中文摘要
我们证明了Riemann zeta函数零点投影纵坐标的对称截断Stieltjes变换的一个无条件Cauchy极限。临界线上zeta对数导数的一个适当归一化具有相同的极限，前提是位于其右侧的零点到临界线的距离之和（按重数计数至高度$2T$）为$o(T)$；我们给出了比较误差的显式界。证明依赖于正平稳点测度的Stieltjes变换的一个收敛定理，其中计数偏差满足一个可积性准则。该定理通过与周期点测度的变换进行比较获得，并结合截断界和一个传递定理。在有限域上，一个经典的余切恒等式给出了簇的zeta函数对数导数的Cauchy极限：对于非恒定的纯上同调因子，该分布恰好是Cauchy分布，而Poincaré对偶为完整的zeta函数提供了显式误差，包括递增次数的光滑超曲面的Cauchy极限，且在基域上一致成立。
英文摘要
We prove an unconditional Cauchy limit for a symmetrically truncated Stieltjes transform of the projected ordinates of the zeros of the Riemann zeta function. A suitable normalization of the logarithmic derivative of $ζ$ on the critical line has the same limit provided that the sum of the distances to the critical line of the zeros lying to its right, counted with multiplicity up to height $2T$, is $o(T)$; we give an explicit bound on the comparison error. The proof rests on a convergence theorem for the Stieltjes transform of positive stationary point measures whose counting discrepancy satisfies an integrability criterion. This theorem is obtained by comparison with the transforms of periodic point measures, combined with truncation bounds and a transfer theorem. Over finite fields, a classical cotangent identity yields Cauchy limits for the logarithmic derivatives of zeta functions of varieties: the law is exactly Cauchy for nonconstant pure cohomological factors, and Poincaré duality gives explicit errors for full zeta functions, including Cauchy limits for smooth hypersurfaces of increasing degree, uniformly in the base field.
CommentsThis paper grows out of the Cauchy-law result for Stieltjes transforms in arXiv:2202.04284, which is here generalized and substantially extended into a self-contained paper
