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Weighted mean squares of ζ and L(·,χ4): 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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Cite

BibTeXRIS
BibTeX
@misc{darses2026weighted,
  title = {{Weighted mean squares of $ζ$ and $L(\cdot,χ_4)$: closed forms, reciprocity, and a missing identity}},
  author = {Sébastien Darses and Erwan Hillion},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.36200},
  eprint = {2609.36200},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36200},
}
RIS
TY  - UNPB
TI  - Weighted mean squares of ζ and L(·,χ_4): closed forms, reciprocity, and a missing identity
AU  - Sébastien Darses
AU  - Erwan Hillion
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.36200
UR  - https://arxiv.org/abs/2609.36200
AB  - 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 $4$.
ER  -