The evaluation of the Riemann zeta function at odd integers remains an important analytical problem, partly because many of its classical series representations converge slowly. In this paper, we develop a continuous filtering framework for analyzing weighted series representations associated with $ \zeta(5) $. Rather than relying on closed-form algebraic transformations, the proposed approach introduces analytically designed weighting sequences (filters) within a continuous integral formulation. We show that this construction naturally interprets discrete filtering as a continuous transformation of the underlying integral kernel. Furthermore, the proposed framework reveals a fundamental trade-off between convergence enhancement and preservation of the kernel structure. By establishing a continuous–discrete connection, the framework provides a unified perspective for studying filtered series representations and offers new insight into the analytical structure of zeta-related series.
Citation: Sunyoung Bu, Jongkyum Kwon. Continuous filtering and kernel transformations in series representations of $ \zeta(5) $[J]. Electronic Research Archive, 2026, 34(11): 8101-8118. doi: 10.3934/era.2026345
The evaluation of the Riemann zeta function at odd integers remains an important analytical problem, partly because many of its classical series representations converge slowly. In this paper, we develop a continuous filtering framework for analyzing weighted series representations associated with $ \zeta(5) $. Rather than relying on closed-form algebraic transformations, the proposed approach introduces analytically designed weighting sequences (filters) within a continuous integral formulation. We show that this construction naturally interprets discrete filtering as a continuous transformation of the underlying integral kernel. Furthermore, the proposed framework reveals a fundamental trade-off between convergence enhancement and preservation of the kernel structure. By establishing a continuous–discrete connection, the framework provides a unified perspective for studying filtered series representations and offers new insight into the analytical structure of zeta-related series.
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Sunyoung Bu, Jongkyum Kwon. Continuous filtering and kernel transformations in series representations of $ \zeta(5) $[J]. Electronic Research Archive, 2026, 34(11): 8101-8118. doi: 10.3934/era.2026345
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