PreprintCC BY 4.0arXiv
On the Imaginary Part of ζ'/ζ
Abstract
We derive an explicit zero-side formula for the imaginary part of a smooth truncation of the logarithmic derivative of the Riemann zeta-function. The resulting kernel is the odd companion of the Fejér kernel in Montgomery's pair-correlation identity. Motivated by the recent breakthrough of Anthropic's Claude \cite{Claude26}, we also give an expository account of how the real-part identity can be turned into a matrix problem, overcoming the clustering obstruction for zeros off the critical line. Finally, we discuss how the new imaginary-part identity may contain additional information about such clustering.
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BibTeX
@misc{aryan2026imaginary,
title = {{On the Imaginary Part of $ζ'/ζ$}},
author = {Farzad Aryan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12853},
eprint = {2610.12853},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12853},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Imaginary Part of ζ'/ζ
AU - Farzad Aryan
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12853
UR - https://arxiv.org/abs/2610.12853
AB - We derive an explicit zero-side formula for the imaginary part of a smooth truncation of the logarithmic derivative of the Riemann zeta-function. The resulting kernel is the odd companion of the Fejér kernel in Montgomery's pair-correlation identity. Motivated by the recent breakthrough of Anthropic's Claude \cite{Claude26}, we also give an expository account of how the real-part identity can be turned into a matrix problem, overcoming the clustering obstruction for zeros off the critical line. Finally, we discuss how the new imaginary-part identity may contain additional information about such clustering.
ER -