Zeta logarithmic derivative zero bound

Given: Countable choice and the data and hypotheses of the statement.

The Riemann xi function has its genus-one Hadamard product over the nontrivial zeros of zeta: There exist constants $A,B\in \mathbb{C}$ such that $\xi (s)=e^{A+Bs}{\prod }_{\rho }{E}_1(s\mathrm{/}\rho ),$ where the product runs over the nontrivial zeros $\rho$ of $\zeta$, counted with multiplicity, and $E_1(w)=(1-w)e^w\mathrm{.}$ The product converges in the genus-one canonical sense.

The Riemann xi function $\xi (s)=\frac{1}{2}s(s-1)\mathrm{\Lambda }(s)$: The completed function extends meromorphically with simple poles at $0$ and $1$ by thm-completed-riemann-zeta-functional-equation. The Riemann xi function is defined on $\mathbb{C}$ by $\xi (s):=\frac{1}{2}s(s-1)\mathrm{\Lambda }(s)=\frac{1}{2}s(s-1){\pi }^{-s\mathrm{/}2}\mathrm{\Gamma }(s\mathrm{/}2)\zeta (s)\mathrm{.}$ The role of the factor $\frac{1}{2}s(s-1)$ is to cancel the two simple poles of the completed function $\mathrm{\Lambda }$. Thus $\xi$ is the entire completion, while $\mathrm{\Lambda }$ remains meromorphic.

Stirling's formula for Gamma: Fix $\delta$ with $0<\delta <\pi$. On the closed sector $\mathrm{\mid }\arg z\mathrm{\mid }\le \pi -\delta$, using the principal logarithm in $z^{z-1\mathrm{/}2}:=\exp ((z-1\mathrm{/}2)\mathrm{Log}z)$, one has $\mathrm{\Gamma }(z)=\sqrt{2\pi }{z}^{z-1\mathrm{/}2}{e}^{-z}(1+O_{\delta }(\mathrm{\mid }z{\mathrm{\mid }}^{-1}))$ as $\mathrm{\mid }z\mathrm{\mid }\to \mathrm{\infty }$.

All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle: Let $f$ be holomorphic on $D(a,R)$, let $0<r<R$, and let $\gamma (t)=a+r\exp (it)$ for $0\le t\le 2\pi$. Define $f^{(0)}=f$ and, whenever it exists, $f^{(n+1)}=(f^{(n)}{)}^{\mathrm{\prime }}$. Then every $f^{(n)}$ exists on $D(a,r)$ and, for every $z\in D(a,r)$ and $n\in \mathbb{N}$, $f^{(n)}(z)=\frac{n!}{2\pi i}{\int }_{\gamma }\frac{f(\zeta )}{(\zeta -z{)}^{n+1}}d\zeta \mathrm{.}$ In particular, every holomorphic function has complex derivatives of all orders locally.

A unit-interval bound for zeta zeros: The number of nontrivial zeta zeros, with multiplicity, whose ordinates lie in $[T,T+1]$ is $O(\log (T+2))$ for $T\ge 0$.

The only zeros of zeta on the nonpositive real axis are the negative even integers, and every other zero lies in the open critical strip: For each integer $m\ge 1$, $\zeta (-2m)=0.$ These are the only zeros of $\zeta$ on the nonpositive real axis. Every other zero $\rho$ of $\zeta$ satisfies $0<\mathrm{Re}\rho <1$. Moreover, if $\rho$ is a nontrivial zero, then so are $1-\rho$ and $\bar{\rho }$.