A local formula for the logarithmic derivative of zeta

Put $s_0=2+it$. Logarithmic differentiation of the Hadamard product and subtraction at $s_0$ give $\frac{{\xi }^{\mathrm{\prime }}}{\xi }(s)-\frac{{\xi }^{\mathrm{\prime }}}{\xi }(s_0)={\sum }_{\rho }(\frac{1}{s-\rho }-\frac{1}{s_0-\rho })\mathrm{.}$ The constants and genus-one correction terms cancel; the difference series converges absolutely, since its terms are $O_s(\mathrm{\mid }\rho {\mathrm{\mid }}^{-2})$ for large $\mathrm{\mid }\rho \mathrm{\mid }$.

For $\mathrm{\mid }t-\mathrm{\Im }\rho \mathrm{\mid }\ge 1$, each difference has absolute value at most $3\mathrm{/}\mathrm{\mid }t-\mathrm{\Im }\rho {\mathrm{\mid }}^2$. By [L1] and [L2], grouping into the bands $k\le \mathrm{\mid }t-\mathrm{\Im }\rho \mathrm{\mid }<k+1$ bounds their total by $C{\sum }_{k\ge 1}\frac{\log (\mathrm{\mid }t\mathrm{\mid }+k+3)}{k^2}=O(\log (\mathrm{\mid }t\mathrm{\mid }+2))\mathrm{.}$ Here $\log (\mathrm{\mid }t\mathrm{\mid }+k+3)\le \log (\mathrm{\mid }t\mathrm{\mid }+2)+\log (k+3)$, and both resulting weighted series converge. For the remaining zeros, $\mathrm{\mid }{s}_0-\rho \mathrm{\mid }\ge 1$, so the sum of their subtracted terms is also $O(\log (\mathrm{\mid }t\mathrm{\mid }+2))$ by [L1] and [L2]. Thus $\frac{{\xi }^{\mathrm{\prime }}}{\xi }(s)={\sum }_{\mathrm{\mid }t-\mathrm{\Im }\rho \mathrm{\mid }<1}\frac{1}{s-\rho }+\frac{{\xi }^{\mathrm{\prime }}}{\xi }(s_0)+O(\log (\mathrm{\mid }t\mathrm{\mid }+2))\mathrm{.}$

By The Riemann xi function $\xi (s)=\frac{1}{2}s(s-1)\mathrm{\Lambda }(s)$ and the Gamma recurrence, $\xi (s)=(s-1){\pi }^{-s\mathrm{/}2}\mathrm{\Gamma }(1+s\mathrm{/}2)\zeta (s)\mathrm{.}$ The Gamma factor has argument with real part at least $1\mathrm{/}2$. Stirling's formula for Gamma, differentiated using Cauchy's estimate on disks of radius proportional to the argument's modulus in a slightly larger sector, gives ${\mathrm{\Gamma }}^{\mathrm{\prime }}(z)\mathrm{/}\mathrm{\Gamma }(z)=\mathrm{Log}z+O(1\mathrm{/}\mathrm{\mid }z\mathrm{\mid })$ there for large $\mathrm{\mid }z\mathrm{\mid }$; compact subsets of this half-plane supply the remaining bound. Also The logarithmic derivative of the zeta Dirichlet series is the Dirichlet series of the von Mangoldt function on Re s greater than 1 gives $\mathrm{\mid }{\zeta }^{\mathrm{\prime }}\mathrm{/}\zeta (2+it)\mathrm{\mid }\le {\sum }_{n\ge 2}(\log n)n^{-2}<\mathrm{\infty }$. Hence ${\xi }^{\mathrm{\prime }}\mathrm{/}\xi (s_0)=O(\log (\mathrm{\mid }t\mathrm{\mid }+2))$. Substituting the displayed xi identity into step 2.1 leaves the pole term $-1\mathrm{/}(s-1)$ and a Gamma logarithmic derivative of size $O(\log (\mathrm{\mid }t\mathrm{\mid }+2))$, proving the stated uniform formula even at bounded ordinates.