Rogers-Ramanujan连分数的对数矩与奇数zeta值
Logarithmic moments of the Rogers Ramanujan continued fraction and odd zeta values
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中文总结 AI 辅助
该研究探讨Rogers-Ramanujan连分数的尖点展开对其普通与对数Mellin矩的控制作用，证明相关系数的代数性质，确定归一化矩生成函数的极点，并建立反射恒等式等结果。
中文摘要 AI 辅助
我们研究Rogers-Ramanujan连分数的两个尖点展开如何控制其普通Mellin矩与对数Mellin矩。Ramanujan的因式分解将对数变换表示为zeta函数与二次Dirichlet L-函数乘积的组合；其相邻整数与半整数值为对应的奇数特殊值提供了可逆坐标变换。我们证明每个倒数尖点系数都是二次无理代数整数，具有显式极小多项式，且每个对数系数均为二次无理数。这些事实确定了三个归一化矩生成函数的精确极点：对数矩生成函数具有绝对收敛的部分分式展开，在每个正平方处有一个极点；普通矩生成函数则通过双尖点不完全伽马展开，在每个正分裂点处给出亚纯延拓。一个通用Mellin引理导出了相应的非全纯性结论，我们还建立了反射恒等式、垂直带衰减性及显式截断估计。
英文摘要
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.
