Paper
On the series expansion of the secondary zeta function about $s=1$ and its coefficients
Authors
Artur Kawalec
Abstract
The secondary zeta function is defined as a generalized zeta series over the imaginary parts of non-trivial zeros assuming (RH). This function admits Laurent series expansion at the double pole at $s=1$. In this article, we derive a new formula for the expansion coefficients of the regular part, which is similar to the Stieltjes constants formula for the Riemann zeta function. We also numerically verify and compute the new formula to high precision for several test cases. Lastly, we also apply the Brent's (BPT) Theorem for improving convergence of the main formula.
Metadata
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Raw Data (Debug)
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.21555v1</id>\n <title>On the series expansion of the secondary zeta function about $s=1$ and its coefficients</title>\n <updated>2026-03-23T04:18:54Z</updated>\n <link href='https://arxiv.org/abs/2603.21555v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.21555v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>The secondary zeta function is defined as a generalized zeta series over the imaginary parts of non-trivial zeros assuming (RH). This function admits Laurent series expansion at the double pole at $s=1$. In this article, we derive a new formula for the expansion coefficients of the regular part, which is similar to the Stieltjes constants formula for the Riemann zeta function. We also numerically verify and compute the new formula to high precision for several test cases. Lastly, we also apply the Brent's (BPT) Theorem for improving convergence of the main formula.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.NT'/>\n <published>2026-03-23T04:18:54Z</published>\n <arxiv:comment>9 pages, 1 table</arxiv:comment>\n <arxiv:primary_category term='math.NT'/>\n <author>\n <name>Artur Kawalec</name>\n </author>\n </entry>"
}