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The Riemann Hypothesis on Young's Lattice

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Contributed by José Damián Espinosa ↗

Version 1 / Oct 21, 2026 / CC BY 4.0

Abstract

This work establishes a novel structural connection between the Riemann Hypothesis and Young's Lattice through partition-indexed harmonic numbers and the Generalized Assembly Theorem. We establish a new family of equivalences for the conjecture, define the partition-based harmonic numbers HnλH_n^{\boldsymbol{\lambda}}, and reformulate the Riemann Hypothesis as the Young's Lattice equivalence: $$\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}} \frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k > \sum_{d \mid n} d$$ for every n∈Nn \in \mathbb{N}, where λ⊢k\boldsymbol{\lambda} \vdash k denotes a partition of kk, λ!\boldsymbol{\lambda}! is the partition factorial, and ak=1k+⋯+1k2a_k = \frac{1}{k} + \dots + \frac{1}{k^2} or ak=log⁡(k+1)a_k=\log(k+1). We then prove the Generalized Assembly Theorem, which shows that the assembly sum $\mathfrak{S}_n^m = \sum_{k=1}^{n} \sum_{\boldsymbol{\lambda} \in \wp_k^{k+m-1}} H_n^{\boldsymbol{\lambda}}$ equals nn for every excess parameter m−1m-1, providing a systematic framework to analyze divisor bounds through the branch decomposition of the partition-indexed series. The resulting structure yields an exact, non-asymptotic realization of the divisors of nn. Finally, we propose and explore a hierarchical expansion connecting the exponential of the Euler--Mascheroni constant, eγe^\gamma, to Multiple Zeta Values under Abel summability. $$e^\gamma = 1 + \sum_{a\ge2} \frac1{a!} \sum_{m\ge0} (-1)^m\zeta^{\mathrm{A}}(a+m) + \sum_{a,b\ge2} \frac1{a!\,b!} \sum_{m,n\ge0} (-1)^{m+n}\zeta^{\mathrm{A}}(a+m,b+n) + \cdots$$

References

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  6. Espinosa, José Damián. A Beautiful Equivalence of the Riemann Hypothesis (The Riemann Hypothesis and Young's Lattice [Part 1/9]). 2025. Zenodo preprintDOI
  7. Espinosa, José Damián. On a Theorem of Equivalences of the Riemann Hypothesis (The Riemann Hypothesis and Young's Lattice [Part 2/9]). 2025. Zenodo preprintDOI
  8. Espinosa, José Damián. Two T(k)\mathfrak{T}(k)-Integral Equivalences of the Riemann Hypothesis (The Riemann Hypothesis and Young's Lattice [Part 3/9]). 2025. Zenodo preprintDOI
  9. Espinosa, José Damián. Two T(k)\mathfrak{T}(k)-Summation Equivalences of the Riemann Hypothesis (The Riemann Hypothesis and Young's Lattice [Part 4/9]). 2025. Zenodo preprintDOI
  10. Espinosa, José Damián. Five Equivalences of the Riemann Hypothesis (The Riemann Hypothesis and Young's Lattice [Part 5/9]). 2025. Zenodo preprintDOI
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  1. Espinosa, José Damián. On the Generalized Harmonic Number of a Partition λ\boldsymbol{\lambda} and a kk-tuple λ\lambda (The Riemann Hypothesis and Young's Lattice [Part 6/9]). 2025. Zenodo preprintDOI
  2. Espinosa, José Damián. The Assembly Theorem (The Riemann Hypothesis and Young's Lattice [Part 7/9]). 2026. Zenodo preprintDOI
  3. Espinosa, José Damián. My Path Toward the Proof of the Riemann Hypothesis (The Riemann Hypothesis and Young's Lattice [Part 8a/9]). 2026. Zenodo preprintDOI
  4. Espinosa, José Damián. The Riemann Hypothesis Young's Lattice Equivalence (The Riemann Hypothesis and Young's Lattice [Part 8b/9]). 2026. Zenodo preprintDOI
  5. Espinosa, José Damián. The Generalized Assembly Theorem (The Riemann Hypothesis and Young's Lattice [Part 8c/9]). 2026. Zenodo preprintDOI
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  11. Sergiu Alexandru Bîsceanu. n,m,s∈Nn, m, s\in\mathbb{N},s≤ns\leq n: ∑k=sn∑λ∈℘k−s+1k−s+mHs,nλ=n−(s−1)sm\sum_{k=s}^{n}\sum_{\boldsymbol{\lambda}\in\wp_{k-s+1}^{k-s+m}}H_{s,n}^{\boldsymbol{\lambda}}=\frac{n-(s-1)}{s^m}?. (version: 2026-07-12). MathOverflowlink

Version history

  1. v1Submitted by José Damián EspinosaInitial depositCurrentOct 20, 2026