Exact Moments and CM Periods of Reciprocal Quadratic Mellin Integrals
Abstract
We study exact moments and algebraic sampling of a reciprocal quadratic Mellin integral. A finite elementary formula determines every nonzero rational-weight first moment with specified branches. For |b|<2√c, mixed half-weight moments admit a dilogarithmic formula, and all even half-weight moments follow from one polynomial recurrence with a geometric error bound. At zero Mellin weight with b=0, half-integer deformations form an explicit two-function elliptic space. At the singular boundary, a two-parameter reflection gives a radical integral equal to π for every parameter, a cubic logarithmic integral in dilogarithms, and an elementary golden-ratio specialization. All odd zero-weight boundary moments also have explicit formulas. For a reduced rational weight p/q, the sampling function generates a field of degree q2 and genus $(q-1)(q-2)/2$, outside the stated degeneracies. Explicit cubic maps and a Picard--Fuchs certificate fix its elliptic period normalization and yield CM companion series with polynomial-weight closure. At denominator four, we determine the three elliptic quotient maps and their differential pullbacks. For the original kernel (c,b)=(5,−2), the Jacobian is isogenous to E1728× E02, and homological transfer gives gamma-valued periods of the sampling differential. Classical identities are credited and used as ingredients in these family-specific formulas.
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BibTeX
@misc{raghava2026exact,
title = {{Exact Moments and CM Periods of Reciprocal Quadratic Mellin Integrals}},
author = {K. Srinivasa Raghava},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20201},
eprint = {2609.20201},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20201},
}
RIS
TY - UNPB
TI - Exact Moments and CM Periods of Reciprocal Quadratic Mellin Integrals
AU - K. Srinivasa Raghava
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.20201
UR - https://arxiv.org/abs/2609.20201
AB - We study exact moments and algebraic sampling of a reciprocal quadratic Mellin integral. A finite elementary formula determines every nonzero rational-weight first moment with specified branches. For |b|<2√c, mixed half-weight moments admit a dilogarithmic formula, and all even half-weight moments follow from one polynomial recurrence with a geometric error bound. At zero Mellin weight with b=0, half-integer deformations form an explicit two-function elliptic space. At the singular boundary, a two-parameter reflection gives a radical integral equal to π for every parameter, a cubic logarithmic integral in dilogarithms, and an elementary golden-ratio specialization. All odd zero-weight boundary moments also have explicit formulas. For a reduced rational weight p/q, the sampling function generates a field of degree q^2 and genus $(q-1)(q-2)/2$, outside the stated degeneracies. Explicit cubic maps and a Picard--Fuchs certificate fix its elliptic period normalization and yield CM companion series with polynomial-weight closure. At denominator four, we determine the three elliptic quotient maps and their differential pullbacks. For the original kernel (c,b)=(5,−2), the Jacobian is isogenous to E_1728× E_0^2, and homological transfer gives gamma-valued periods of the sampling differential. Classical identities are credited and used as ingredients in these family-specific formulas.
ER -