Riemann and the logarithmic derivatives of zeta
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913165626834944 |
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| author | de Reyna, J. Arias |
| author_facet | de Reyna, J. Arias |
| contents | 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 \[\frac{ζ'(\frac12)}{ζ(\frac12)}=\fracπ{4}+\fracγ{2}+\frac{\log(8π)}{2},\quad \frac{ζ''(\frac12)}{ζ(\frac12)}-\Bigl(\frac{ζ'(\frac12)}{ζ(\frac12)}\Bigr)^2=8-\frac{π^2}{4}-2G+2\sum_{n=1}^\infty\frac{1}{α_n^2}\] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27552 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Riemann and the logarithmic derivatives of zeta de Reyna, J. Arias History and Overview Number Theory Primary 11M06, Secondary 01A55 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 \[\frac{ζ'(\frac12)}{ζ(\frac12)}=\fracπ{4}+\fracγ{2}+\frac{\log(8π)}{2},\quad \frac{ζ''(\frac12)}{ζ(\frac12)}-\Bigl(\frac{ζ'(\frac12)}{ζ(\frac12)}\Bigr)^2=8-\frac{π^2}{4}-2G+2\sum_{n=1}^\infty\frac{1}{α_n^2}\] |
| title | Riemann and the logarithmic derivatives of zeta |
| topic | History and Overview Number Theory Primary 11M06, Secondary 01A55 |
| url | https://arxiv.org/abs/2605.27552 |