Slip Certificates for the Riemann ξ–Function via Poisson Forcing and Carleson Tents a Stagewise One-Dimensional Criterion for Zero-Free Rectangles
Let $xi(s)$ be the completed Riemann zeta function and $Xi(z)=xi(tfrac12+ii z)$. We study the logarithmic derivative field [ m(z):=-frac{Xi'(z)}{Xi(z)} ] and introduce a one-dimensional functional along horizontal scan lines $z=t+iieta$: for a bounded interval $IsubsetRR$ and $eta>0$, [ Slip^+_{eta}(I):=int_I pos{-Ima m(t+iieta)},dd t, ] with $Slip^+_{eta}(I)=+infty$ if $Xi(t+iieta)=0$ for some $tin I$. Writing $s=tfrac12+ii z=(tfrac12-eta)+ii t$, one has [ -Ima m(t+iieta)=Rea!left(frac{xi’}{xi}(s)right) =frac{partial}{partialsigma}log|xi(sigma+ii t)| qquad(sigma=tfrac12-eta), ] so $-Ima m$ is a vertical derivative of $log|xi|$.Our first main result is a […]
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