Paper
On the series expansion of the prime zeta function about $s=1$ and its coefficients
Authors
Artur Kawalec
Abstract
In this article, we derive a series expansion of the prime zeta function about the $s=1$ logarithmic singularity and prove general formula for its expansion coefficients, which is similar to the Stieltjes expansion coefficients for the Riemann zeta function. These results can also be viewed as a generalization of Mertens's Theorems to higher order. We also numerically verify and compute the presented formulas to high precision for several test cases.
Metadata
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.21535v1</id>\n <title>On the series expansion of the prime zeta function about $s=1$ and its coefficients</title>\n <updated>2026-03-23T03:44:52Z</updated>\n <link href='https://arxiv.org/abs/2603.21535v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.21535v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>In this article, we derive a series expansion of the prime zeta function about the $s=1$ logarithmic singularity and prove general formula for its expansion coefficients, which is similar to the Stieltjes expansion coefficients for the Riemann zeta function. These results can also be viewed as a generalization of Mertens's Theorems to higher order. We also numerically verify and compute the presented formulas to high precision for several test cases.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.NT'/>\n <published>2026-03-23T03:44:52Z</published>\n <arxiv:comment>1 Table</arxiv:comment>\n <arxiv:primary_category term='math.NT'/>\n <author>\n <name>Artur Kawalec</name>\n </author>\n </entry>"
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