[论文解读] The Riemann Hypothesis in Oaxaca

Carlos Segovia|arXiv (Cornell University)|Jan 29, 2026

Analytic Number Theory Research被引用 0

一句话总结

论文通过分区与 Young 晓 lattice 的关系，将 RH 等价性与 A(n) 的渐近性质联系起来，给出 r≥1 的显式极限比例 (rho_r)，并讨论超越简单 r,1^l 分区的必要校正项。

ABSTRACT

An equivalence of the Riemann Hypothesis (RH) enables a direct bridge to the Young lattice. In specific, the classical threshold $\lim_{n o\infty} σ(n)/(n \log\log n) = e^γ \approx 1.78107$, derived from the asymptotic behavior of the sum-of-divisors function, can be realized combinatorially via limiting proportions associated to specific families of integer partitions.

研究动机与目标

• Motivate the RH problem and its classical equivalences with divisor-sum bounds.
• Embed these equivalences in combinatorial terms via integer partitions and the Young lattice.
• Derive asymptotic behavior of A(n) and its relation to e^γ n log log n.
• Identify and discuss necessary correction terms beyond simple partition structures to match RH thresholds.

提出的方法

• Review of classical RH equivalences (Robin and Lagarias) and their inequalities.
• Definition of A(n) as a weighted sum of (H_n)^k/k! and its relation to σ(n).
• Application of partition theory and monomial/e elementary symmetric polynomials to express A(n) (Espinosa's assembly theorem).
• Derivation of asymptotic limits ρ_r for A_r(n)/ (n log log n) and closed forms involving zeta values.
• Use of central moments and cumulants of the number of cycles of random permutations to estimate E_j terms.
• Discussion of correction terms R_r(n) arising from partitions beyond [r,1^l].

实验结果

研究问题

• RQ1 Can RH be characterized by inequalities involving σ(n) and n log log n within the combinatorial framework of partitions?
• RQ2 What are the limiting proportions ρ_r contributing to the RH threshold when A(n) is decomposed across partition types?
• RQ3 How do non-[r,1^l] partitions (e.g., [2,2], [3,2], [2,2,2]) affect the RH-related bounds and the convergence to e^γ?
• RQ4 What is the role of the asymptotic behavior of A(n) in relation to the Young lattice and assemblies of symmetric functions?

主要发现

• The classical threshold lim_{n→∞} σ(n)/(n log log n) = e^γ provides a benchmark linked to RH.
• The derived limits ρ_r give a decreasing sequence with ρ_1 = 1 and ρ_2 = 0.5, approaching smaller values for higher r.
• A closed-form-like expression ρ_r = (1/r!) [(-1)^r + ∑_{j=3}^r (-1)^{r+j} ζ(j-1)] is obtained for r ≥ 3.
• A table of finite-r contributions shows cumulative sums S_r converging to about 1.6359, highlighting the need for corrections beyond simple partitions to reach e^γ (≈1.7810).
• An explicit correction term R_3(n) is provided for the [2,2,1^l] partitions, illustrating how non-[r,1^l] partitions influence the RH-bound arguments.
• Computational notes indicate the proportion between A_r(n) and R_r(n) grows initially, then behaviorally decreases, emphasizing when and how corrections become relevant.
• The work connects divisor-sum inequalities to probabilistic interpretations via cycle counts of random permutations and related symmetric function theory.