Riemann and the logarithmic derivatives of zeta

Abstract

In one of his posthumous papers, conserved in Göttingen, Riemann considers the derivatives of $\logζ(s)$ at the point $1/2$, giving explicit values for them. Around 2010 we shared Riemann's value of the second derivative with some mathematicians. From that time I have been asked several times for references. So I decided to write this. Specially explaining the wonderful formulas \[\frac{ζ'(\frac12)}{ζ(\frac12)}=\fracπ{4}+\fracγ{2}+\frac{\log(8π)}{2},\quad \frac{ζ''(\frac12)}{ζ(\frac12)}-\Bigl(\frac{ζ'(\frac12)}{ζ(\frac12)}\Bigr)^2=8-\frac{π^2}{4}-2G+2\sum_{n=1}^\infty\frac{1}{α_n^2}\]