Trigonometric Selector Kernels, Duality, and Odd Zeta Values

Ken Nagai

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arXiv · 2509.10801

Trigonometric Selector Kernels, Duality, and Odd Zeta Values

Ken Nagai

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Abstract

In this short note, we develop trigonometric selector kernels to represent odd zeta values via dual hyperbolic counterparts. This framework highlights a Fourier-Poisson duality, incorporating finite-part integrals in the sense of Hadamard-Galapon. In particular, we show how such kernels naturally recover Euler-Maclaurin and Poisson summation formulas as dual manifestations. We further connect our kernel approach with the finite-part integral formulation, extending earlier Cvijovi\'c-Klinowski type representations for odd zeta values.

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Ken Nagai. 2025-09-13. Trigonometric Selector Kernels, Duality, and Odd Zeta Values. https://arxiv.org/abs/2509.10801

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