Abstract:In one of his posthumous papers, conserved in Göttingen, Riemann considers the derivatives of $\log\zeta(s)$ 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 \[\frac{\zeta'(\frac12)}{\zeta(\frac12)}=\frac{\pi}{4}+\frac{\gamma}{2}+\frac{\log(8\pi)}{2},\quad \frac{\zeta''(\frac12)}{\zeta(\frac12)}-\Bigl(\frac{\zeta'(\frac12)}{\zeta(\frac12)}\Bigr)^2=8-\frac{\pi^2}{4}-2G+2\sum_{n=1}^\infty\frac{1}{\alpha_n^2}\]
From: Juan Arias De Reyna [view email]
[v1]
Tue, 26 May 2026 18:18:47 UTC (2,797 KB)
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