On the Generalized Harmonic Number of a Partition $\mathbf{\lambda}$ and a $k$-tuple $\lambda$ (The Riemann Hypothesis and Young´s Lattice [Part 6/9])

On the Generalized Harmonic Number of a Partition λand a k-tuple λ José Damián Espinosa December 21, 2025 Dedication: Στην αγαπημένη μου Θεά Μαρσέλα (A mi amada Diosa Marcela) Abstract This article proposes for n∈N, the n-th generalized harmonic number of order λ1+λ2+. . .+λkassociated with the partition λ= (λ1, λ2,...,λk): Hλ n=X Θ∈Ψ(λ) X 1≤i1<i2<···<ik≤n k Y r=1 1 iσr r ; the n-th generalized harmonic number of order λ1+λ2+...+λkassociated with the k-tuple λ= (λ1, λ2,...,λk): Hλ n=X 1≤i1<i2<···<ik≤n k Y r=1 1 iλr r ; it also proposes the relationship between them: Hλ n=X Θ∈Ψ(λ) HΘ n, and using all this, writes Hk n k!in this unique and magnificent, so powerful and useful form: Hk n k!=X λ⊢k Hλ n λ!=X λ⊢k 1 λ!X Θ∈Ψ(λ) HΘ n. 1 “The essence of mathematics is not to make simple things complicated, but to make complicated things simple.”– S. Gudder “Everything should be made as simple as possible, but not simpler.”– Albert Einstein “Truth is ever to be found in simplicity, and not in the multiplicity and confusion of things.”– Isaac Newton “Simplicity is the ultimate sophistication.”– Leonardo da Vinci “Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without any appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a strict perfection such as only the greatest art can show.”– Bertrand Russell “Beauty is the first test: there is no permanent place in the world for ugly mathematics.”– G. H. Hardy “A mathematician is not complete until he is a bit of a poet in his soul.”– Sofia Kovalevskaya “The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.”– Henri Poincaré “Mathematics possesses not only truth, but supreme beauty.”– Bertrand Russell 2 “Imagination is more important than knowledge. Knowledge is limited. Imagination encircles the world.”– Albert Einstein “Logic will get you from A to B. Imagination will take you everywhere.”– Albert Einstein “If I had an hour to solve a problem, I’d spend 55 minutes thinking about the problem and 5 minutes thinking about solutions.”– Albert Einstein “An expert is a person who has made all the mistakes that can be made in a very narrow field.”– Niels Bohr “To achieve the impossible, you must attempt the absurd.”– Miguel de Cervantes “We must know, we will know ( Wir müssen wissen. Wir werden wissen.).”– David Hilbert Contents 1 Introduction 4 2 Content 5 2.1 On the Generalized Harmonic Number of a Partition λand a k-tuple λ............................... 5 2.1.1 Preliminary Concepts . . . . . . . . . . . . . . . . . . . . 5 2.1.2 About Hλ nand Hλ n...................... 5 2.1.3 On Hk n k!............................. 8 2.1.4 A Partition Decomposition of the Multinomial Theorem . 11 References 13 3 1 Introduction Harmonic numbers and their various generalizations have been a cornerstone of number theory, combinatorics, and classical analysis since the seminal works of Euler. While the classical n-th harmonic number of order mis well-understood as H(m) n=Pn k=1 1 km, the internal structure of their powers and the underlying connection to multiple reciprocal sums reveal a far more complex algebraic landscape. These structures are intimately linked to multiple zeta values (MZVs) and have become increasingly vital in modern theoretical physics, particularly in the evaluation of Feynman integrals and scattering amplitudes. The primary mathematical challenge in this field resides in the expansion of the k-th power of a harmonic sum, Hk n. Using traditional algebraic methods, such expansion leads to a labyrinth of cross-terms and indices that lack a clear combinatorial organization as kincreases. This article addresses this complexity by proposing a robust framework for the n-th generalized harmonic number of order λ1+λ2+· · ·+λkassociated with the partition λ= (λ1, λ2, . . . , λk), defined as: Hλ n=X Θ∈Ψ(λ)X 1≤i1<i2<···<ik≤n k Y r=1 1 iσr r . Furthermore, we define n-th generalized harmonic number of order λ1+λ2+ · · · +λkassociated with the specific k-tuple λ= (λ1, λ2, . . . , λk): Hλ n=X 1≤i1<i2<···<ik≤n k Y r=1 1 iλr r . By establishing the fundamental relationship Hλ n=PΘ∈Ψ(λ)HΘ n, this paper replaces brute-force expansion with a structured decomposition based on integer partitions. This approach reveals that the power of a harmonic number is not merely a repeated product, but a sum of symmetric structures governed by the symmetry of its underlying partitions λ⊢k. Using this symmetry, we can write the normalized power Hk n k!in this unique and magnificent form: Hk n k!=X λ⊢k Hλ n λ!=X λ⊢k 1 λ!X Θ∈Ψ(λ) HΘ n. This formulation provides an elegant and highly useful tool for the symbolic and computational manipulation of harmonic powers, offering a powerful bridge between the theory of partitions and multiple harmonic sums. The primary motivation for this framework is the inherent complexity of expanding the power of a harmonic sum, Hk n, using traditional algebraic methods. As kincreases, the expansion becomes a labyrinth of cross-terms and indices that lack a clear combinatorial organization. By mapping these powers onto the set of integer partitions λ⊢k, our approach replaces brute-force expansion with a structured decomposition. This transformation reveals that the power of a harmonic 4 number is not merely a repeated product, but a sum of symmetric structures governed by the symmetry of its underlying partitions. Consequently, we can represent the normalized power Hk n k!in this unique and magnificent form. 2 Content 2.1 On the Generalized Harmonic Number of a Partition λand a k-tuple λ In this Section 2.1, we work with a continuous interweaving of results. This set is presented through ordered sections that build upon one another as the narrative progresses, constructively. 2.1.1 Preliminary Concepts This Section 2.1.1 addresses key concepts, which lay the foundation for everything undertaken, for what is constructed later. To establish the combinatorial foundation of our main result, we first recall the Multinomial Theorem. This theorem provides the necessary framework for expanding powers of sums and serves as the bridge between the algebraic product Hk nand the structured partitions of the integer k. In this context, we consider the expansion of a sum of nvariables raised to the power k, which naturally leads to the inclusion of multinomial coefficients and the subsequent organization by partitions. Theorem 1 (Multinomial Theorem).For m∈Nand real numbers x1, . . . , xn, (x1+x2+· · · +xn)m=X α1,...,αn≥0 α1+···+αn=m m α1, α2, . . . , αnxα1 1xα2 2···xαn n where: •α1, . . . , αn∈N; •m α1,α2,...,αn=m! α1!α2!···αn!is the multinomial coefficient. 2.1.2 About Hλ nand Hλ n We begin by establishing the formal language of partitions, which will allow us to categorize the harmonic sums based on the decomposition of their total order. Definition 1 (Partition of an integer).Let m∈N. A partition λof mof length kis a k-tuple of λi∈Nfor 1≤i≤k: λ= (λ1, λ2, . . . , λk)with λ1≥λ2≥···≥λk≥1, k X r=1 λr=m. 5 We write λ⊢mto denote that λis a partition of m. Once the partition is defined, it is necessary to introduce the algebraic weighting associated with it. The following definitions for the factorial and the multinomial coefficient of a partition are essential for the normalization of our final formula. Definition 2 (Partition factorial).For a partition λ= (λ1, . . . , λk)⊢m, we define the partition factorial: λ!:=λ1!λ2!···λk!. Definition 3 (Multinomial coefficient for partitions).For a partition λ= (λ1, . . . , λk)⊢m, we define the partition multinomial coefficient: m λ:=m λ1, . . . , λk=m! λ1!λ2!···λk!=m! λ!. A crucial aspect of our proposal is the distinction between a partition (where order is fixed) and the set of its possible arrangements. To capture the full symmetry of the harmonic power, we define the set of all distinct permutations as follows: Definition 4 (Set of distinct permutations).For a partition λ= (λ1, . . . , λk)⊢ m, let Ψ(λ)denote the set of all distinct rearrangements of λ(the set of all distinct permutations of the k-tuple (λ1, . . . , λk)): Ψ(λ) = n(σ1, σ2, . . . , σk)∈Nk:∀j∈N,card{r:σr=j}= card{r:λr=j}o. Each Θ=(σ1, σ2, . . . , σk)∈Ψ(λ)is a k-tuple that represents a distinct rearrangement of the parts of λ. With these tools, we can now introduce the two core objects of this study. The first definition (Definition 5) provides a global view based on the partition’s symmetry, while the second (Definition 6) focuses on the specific order of the exponents. Definition 5 (Generalized Harmonic Number of a Partition (Form with distinct rearrangements)).Let λ= (λ1, λ2, . . . , λk)be a partition of length kwith λ1≥ λ2≥···≥λk≥1, a k-tuple of exponents with λi∈Nfor 1≤i≤k. For n∈N with n≥k, the n-th generalized harmonic number of order λ1+λ2+. . . +λk associated with the partition λis defined as: Hλ n:=X Θ∈Ψ(λ)X 1≤i1<i2<···<ik≤n k Y r=1 1 iσr r where Ψ(λ)is the set of all distinct rearrangements of λ: Ψ(λ) = n(σ1, σ2, . . . , σk)∈Nk:∀j∈N,card{r:σr=j}= card{r:λr=j}o. Each Θ=(σ1, σ2, . . . , σk)∈Ψ(λ)is a k-tuple that represents a distinct rearrangement of the parts of λ. For n∈Nwith n<k, we define Hλ n:= 0. 6 Definition 6 (Generalized Harmonic Number for a k-tuple (Specific Exponent Partition Assignment)).Let λ= (λ1, λ2, . . . , λk)be a k-tuple of exponents with λi∈Nfor 1≤i≤k. For n∈Nwith n≥k, the n-th generalized harmonic number of order λ1+λ2+. . . +λkassociated with the k-tuple λis defined as: Hλ n:=X 1≤i1<i2<···<ik≤n k Y r=1 1 iλr r . In this definition, the order of the exponents matters: different orderings of the same multiset of exponents generally yield different harmonic numbers. For n∈Nwith n<k, we define Hλ n:= 0. The elegance of this framework lies in the direct correspondence between these two definitions. As shown in Proposition 1, the harmonic number of a partition can be understood as the sum of the harmonic numbers of all its distinct tuple arrangements. Proposition 1 (Relation between the two definitions, Hλ nand Hλ n).For any partition λ= (λ1, λ2, . . . , λk)of length kwith λ1≥λ2≥ · · · ≥ λk≥1, being a k-tuple of exponents with λi∈Nand n∈Nthen: Hλ n=X Θ∈Ψ(λ) HΘ n where: •Ψ(λ)is the set of distinct rearrangements of λas defined above; •Hλ nis the n-th generalized harmonic number of order λ1+λ2+. . . +λk associated with the partition λ; •Hλ nis the n-th generalized harmonic number of order λ1+λ2+. . . +λk associated with the k-tuple λ. Proof. By Definition 5, when n≥k: Hλ n=X Θ∈Ψ(λ)X 1≤i1<i2<···<ik≤n k Y r=1 1 iσr r for each fixed Θ=(σ1, . . . , σk)∈Ψ(λ), but by Definition 6 we have: HΘ n=X 1≤i1<i2<···<ik≤n k Y r=1 1 iλr r , substituting this for all Θ∈Ψ(λ)into the expression of Hλ ngives: Hλ n=X Θ∈Ψ(λ) HΘ n, if n<kthe result is trivial: Hλ n=0=PΘ∈Ψ(λ)0 = PΘ∈Ψ(λ)HΘ n. 7 2.1.3 On Hk n k! Building upon the definitions established in the previous section, we now present the central result of this paper: the decomposition of the power of a harmonic number into a weighted sum of partition-based harmonic numbers. This first representation emphasizes the natural normalization of the harmonic power Hk n k!. Proposition 2 (Partition expansion of the harmonic power I).For all n, k ∈N, Hk n k!=X λ⊢k Hλ n λ!. where: •Hn=Pn i=1 1 iis the n-th harmonic number; •λ⊢kdenotes that λis a partition of k; •Hλ nis the n-th generalized harmonic number of order massociated with the partition λ⊢k; •λ! = λ1!λ2!···λm!is the partition factorial associated with the partition λ⊢k. Proof. We start from the multinomial expansion of the Multinomial Theorem (Theorem 1): Hk n=  n X i=1 1 i  k =X α1,...,αn≥0 α1+···+αn=k k! α1!···αn! n Y j=1 1 jαj. Fix a partition λ= (λ1, . . . , λm)⊢kwith m≤n. Consider the terms in the above sum for which exactly mindices jsatisfy αj>0and the multiset of these positive αjequals {λ1, . . . , λm}. Let 1≤j1<· · · < jm≤nbe those indices. Each assignment of the multiplicities λ1, . . . , λmto the positions j1, . . . , jmcorresponds to a permutation Θ=(σ1, . . . , σm)∈Ψ(λ). For a fixed Θand fixed indices j1<· · · < jm, the corresponding term is: k! σ1!···σm!·1 jσ1 1. . . jσm m . Since Θis just a rearrangement of λ, we have σ1!···σm! = λ1!···λm! = λ!. Thus the term becomes: k! λ!·1 jσ1 1. . . jσm m . Summing over all distinct permutations Θ∈Ψ(λ)and over all strictly increasing m-tuples j1<· · · < jm, we obtain the contribution of the partition λ: X 1≤j1<···<jm≤nX Θ∈Ψ(λ) k! λ! m Y r=1 1 jσr r =k! λ!X 1≤j1<···<jm≤nX Θ∈Ψ(λ) m Y r=1 1 jσr r . 8 By Definition 5, the double sum on the right-hand side is precisely Hλ nand for n<m,Hλ n= 0 (by the same definition). Summing over all partitions λ⊢k gives: Hk n=X λ⊢k k! λ!Hλ n. Finally, dividing both sides by k!yields the desired identity: Hk n k!=X λ⊢k Hλ n λ!. By substituting the relationship between partition-based and tuple-based harmonic numbers, we obtain a more granular expansion. This second form explicitly shows how every distinct permutation of the partition contributes to the total power. Proposition 3 (Partition expansion of the harmonic power II).For all n, k ∈ N, Hk n k!=X λ⊢k 1 λ!X Θ∈Ψ(λ) HΘ n. where: •Hn=Pn i=1 1 iis the n-th harmonic number; •λ⊢kdenotes that λis a partition of k; •Ψ(λ)denotes the set of all distinct rearrangements of λ; •HΘ nis the n-th generalized harmonic number of order kassociated with the m-tuple Θ∈Ψ(λ); •λ! = λ1!λ2!···λm!is the partition factorial associated with the partition λ⊢k. Proof. By Proposition 2: Hk n k!=X λ⊢k Hλ n λ! but by Proposition 1 we have: Hλ n=X Θ∈Ψ(λ) HΘ n. Substituting this into the expression of Hλ ngives: Hk n k!=X λ⊢k 1 λ!X Θ∈Ψ(λ) HΘ n. 9