Riemann and the logarithmic derivatives of zeta

2026/05/26 by J. Arias de Reyna · 1 voice
Mathematics · #math.HO #math.NT

paper · pdf

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 (ζ'(\frac12))/(ζ(\frac12))=\fracπ4+\fracγ2+(log(8π))/(2), (ζ''(\frac12))/(ζ(\frac12))-((ζ'(\frac12))/(ζ(\frac12)))²=8-(π²)/(4)-2G+2∑_n=1^∞(1)/(α_n²)

Citations

• Computation of the secondary zeta function
• The Mangoldt function and the non-trivial zeros of the Riemann zeta function

Discussions

• J. Arias de Reyna: Riemann and the logarithmic derivatives of zeta https://arxiv.org/abs/2605.27552 https://arxiv.org/pdf/2605.27552 https://arxiv.org/html/2605.27552 [bsky, 0 points, 0 comments]

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