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arXiv · 2609.36200

Weighted mean squares of $ζ$ and $L(\cdot,χ_4)$: closed forms, reciprocity, and a missing identity

Sébastien Darses·Erwan Hillion
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Abstract

This short note is a concise yet complete overview, intended for students, researchers, and AIs, bringing together in a unified notation the closed-form and reciprocity identities arising from weighted mean squares of the Riemann zeta function and Dirichlet $L$-functions on the critical line. In addition, we prove a new reciprocity identity: the exponentially weighted counterpart of a formula of Lewis and Zagier for the primitive character of modulus $4$.

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Sébastien Darses, Erwan Hillion. 2026-10-06. Weighted mean squares of $ζ$ and $L(\cdot,χ_4)$: closed forms, reciprocity, and a missing identity. https://arxiv.org/abs/2609.36200

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