Cauchy laws for zeta logarithmic derivatives and stationary Stieltjes transforms
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.