Logarithmic moments of the Rogers-Ramanujan continued fraction and odd zeta values
Abstract
We study how the two cusp expansions of the Rogers-Ramanujan continued fraction control its ordinary and logarithmic Mellin moments. Ramanujan's factorizations express the logarithmic transform as a combination of products of zeta and quadratic Dirichlet L-functions. Its adjacent integer and half-integer values give an invertible change of coordinates for the corresponding odd special values. We prove that every reciprocal-cusp coefficient is a quadratic irrational algebraic integer, with an explicit minimal polynomial, and that every logarithmic coefficient is quadratic irrational. These facts determine the exact poles of three normalized-moment generating functions. The logarithmic one has an absolutely convergent partial fraction expansion with a pole at every positive square. For the ordinary moments, a two-cusp incomplete-gamma expansion gives meromorphic continuation at every positive splitting point. A general Mellin lemma yields the corresponding nonholonomicity statements. We also establish reflection identities, vertical strip decay and an explicit truncation estimate.
Related
- Eta-Quotient Representations for a Three Parameter Family of Modular Functions Associated with the Rogers-Ramanujan Continued Fraction
- A Four-Genus Kronecker-Limit Evaluation of the Alternating Rogers-Ramanujan Continued Fraction
- On Ramanujan's cubic continued fraction, II
- Combinatorial decompositions of strict partitions and multi-sum Rogers-Ramanujan type identities
- An elementary treatment of Ramanujan's elementary continued fractions
BibTeX
@misc{raghava2026logarithmic,
title = {{Logarithmic moments of the Rogers-Ramanujan continued fraction and odd zeta values}},
author = {K. Srinivasa Raghava},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10578},
eprint = {2610.10578},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10578},
}
RIS
TY - UNPB
TI - Logarithmic moments of the Rogers-Ramanujan continued fraction and odd zeta values
AU - K. Srinivasa Raghava
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.10578
UR - https://arxiv.org/abs/2610.10578
AB - We study how the two cusp expansions of the Rogers-Ramanujan continued fraction control its ordinary and logarithmic Mellin moments. Ramanujan's factorizations express the logarithmic transform as a combination of products of zeta and quadratic Dirichlet L-functions. Its adjacent integer and half-integer values give an invertible change of coordinates for the corresponding odd special values. We prove that every reciprocal-cusp coefficient is a quadratic irrational algebraic integer, with an explicit minimal polynomial, and that every logarithmic coefficient is quadratic irrational. These facts determine the exact poles of three normalized-moment generating functions. The logarithmic one has an absolutely convergent partial fraction expansion with a pole at every positive square. For the ordinary moments, a two-cusp incomplete-gamma expansion gives meromorphic continuation at every positive splitting point. A general Mellin lemma yields the corresponding nonholonomicity statements. We also establish reflection identities, vertical strip decay and an explicit truncation estimate.
ER -