On a Theorem of Equivalences of the Riemann Hypothesis (The Riemann Hypothesis and Young´s Lattice [Part 2/9])

On a Theorem of Equivalences of the Riemann Hypothesis José Damián Espinosa December 5, 2025 Dedication: Στην αγαπημένη μου Θεά Μαρσέλα (A mi amada Diosa Marcela) Abstract In this article, a Theorem of Equivalences of the Riemann Hypothesis is presented, for the construction of new elementary versions of the Riemann Hypothesis of the following form: The Riemann Hypothesis is equivalent to: O(Hn) = X k∈N Hk n k!T(k)> σ(n) for all n∈N, where Hn=Pn i=1 1 iis the n-th harmonic number, σ(n) = Pd|ndis the sum-of-divisors function of nand the functions O:R+→R and T:R+→Rsatisfy three certain conditions. “The essence of mathematics is not to make simple things complicated, but to make complicated things simple.”– S. Gudder 1 “Everything should be made as simple as possible, but no 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 a sculpture, without any appeal to our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern 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 little 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, rightly viewed, possesses not only truth, but supreme beauty.”– Bertrand Russell “Imagination is more important than knowledge. Knowledge is limited. Imagination encircles the world.”– Albert Einstein 2 “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 attain the impossible, one must attempt the absurd.”– Miguel de Cervantes “We must know, we will know (Wir müssen wissen. Wir werden wissen.).”– David Hilbert Contents 1 Introduction 3 2 Content 4 2.1 New Theorem of Equivalences of the Riemann Hypothesis . . . . 4 2.1.1 Preliminary Concepts . . . . . . . . . . . . . . . . . . . . 4 2.1.2 On Pk∈N(Hk−γ)xk k!..................... 5 2.1.3 On elementary versions of the Riemann Hypothesis by Lagarias (2002) and Robin (1984) . . . . . . . . . . . . . 5 2.1.4 On the Espinosa-Riemann Equivalence Theorem . . . . . 7 References 13 1 Introduction This brief article arises as a consequence of the work A Beautiful Equivalence of the Riemann Hypothesis, what is proposed here is a brief generalization of the method used for its obtention. 3 2 Content 2.1 New Theorem of Equivalences of the Riemann Hypothesis In this Section 2.1, we work with a continuous interlacing of results until reaching Theorem 4. That set is presented by ordered sections that build upon one another as the narrative progresses, constructively, following the clear natural line that finally flows into the long-awaited mentioned result. 2.1.1 Preliminary Concepts The present Section 2.1.1 addresses key concepts that provide the foundation for everything undertaken, for what is subsequently constructed. It begins by defining the famous Euler-Mascheroni Constant γ, an essential constant when addressing the Riemann Hypothesis, the definition follows: Definition 1 (Euler-Mascheroni Constant γ).The Euler-Mascheroni constant γis defined by the limit (Abramowitz [AS64, Formula 6.1.3]): γ:= lim n→∞   n X k=1 1 k−log n = 0.5772156649 ... This limit exists and is finite, representing the asymptotic difference between the harmonic series and the natural logarithm. Having established these results, we proceed to mention and define some fundamental concepts, such as the convergence and radius of convergence of real series, as follows in Definition 2: Definition 2 (Convergence of real power series).Let P∞ k=0 ckxkbe a power series with ck, x ∈R. We say that (Rudin [Rud64, Definition 3.38]): •The series converges at x∈Rif limn→∞ Pn k=0 ckxkexists and is finite. •The radius of convergence R∈[0,+∞]is the supremum of the |x|for which the series converges. Similarly, the concept of Asymptotic Upper Bound for a real function is defined through Definition 3: Definition 3 (ONotation (Asymptotic Upper Bound)).Let f, g :N→R(or f, g :R→R). We say that: f(x) = O(g(x)) when x→ ∞, if there exist constants x0>0and M > 0such that: |f(x)| ≤ M|g(x)|for all x≥x0. 4 And its fundamental properties as follows in Proposition 1: Proposition 1 (Fundamental Properties of ONotation).Let E⊆Rand let f1, f2, g1, g2:E→Rbe functions. For k∈R\ {0}, the following hold: (T) Transitivity:f1=O(g1)∧g1=O(g2) =⇒f1=O(g2); (M) Multiplication by Function (set equality): f2· O(g1)=O(f2g1); (S) Sum:f1=O(g1)∧f2=O(g2) =⇒f1+f2=O(max(|g1|,|g2|)); (H) Homogeneity:f1=O(g1) =⇒kf1=O(g1). 2.1.2 On Pk∈N(Hk−γ)xk k! Two key results are presented in the work. The series Pk∈N(Hk−γ)xk k!, with two tight bounds, one upper and one lower respectively, and an asymptotic expression in the following two propositions respectively: Proposition 2. Let x∈R+, it holds that: X k∈N (Hk−γ)xk k!=exlog x+γ+O1 xwhen x→ ∞, where: •N={1,2,3, . . .}denotes the set of natural numbers; •Hk=Pk n=1 1 nis the k-th harmonic number (H0:= 0); •γis the Euler-Mascheroni constant. Proof. See in Espinosa [Esp25] (A Beautiful Equivalence of the Riemann Hypothesis). Proposition 3. For x∈R+, with x= 0, the following triple inequality holds: exlog x+1 2log 1 + 2 x+γ < X k∈N (Hk−γ)xk k!< exlog x+ log 1 + 1 x+γ. Proof. See in Espinosa [Esp25] (A Beautiful Equivalence of the Riemann Hypothesis). 2.1.3 On elementary versions of the Riemann Hypothesis by Lagarias (2002) and Robin (1984) Four vital results for this work are stated in this Section 2.1.3. They serve as a bridge between what is constructed here and what has already been done, between the present work and the Riemann Hypothesis. 5 Theorem 1 (Lagarias, 2002).The Riemann Hypothesis is equivalent to the following elementary statement: σ(n)≤Hn+eHnlog(Hn)for all n∈N,(1) where: •σ(n) = Pd|ndis the sum-of-divisors function of n; •Hn=Pn i=1 1 iis the n-th harmonic number. Corollary 1 (Lagarias, 2002).The Riemann Hypothesis is equivalent to the following elementary statement: σ(n)< eHnlog(Hn)for all n∈N,with n>60,(2) where: •σ(n) = Pd|ndis the sum-of-divisors function of n; •Hn=Pn i=1 1 iis the n-th harmonic number. Theorem 2 (Robin, 1984).If the Riemann Hypothesis is: 1. true, then for all integers n≥5041, the following inequality holds: σ(n)< eγnlog log n, (3) where: •σ(n) = Pd|ndis the sum-of-divisors function of n; •γis the Euler-Mascheroni constant (γ≈0.5772). 2. false, then the preceding inequality fails for infinitely many values of n. Theorem 3 (Robin, 1984).If the Riemann Hypothesis is false, then there exist constants C > 0and β(with 0< β < 1 2) such that: σ(n)≥eγnlog log n+C n log log n (log n)βfor infinitely many values of n, (4) with n∈N, where: •σ(n) = Pd|ndis the sum-of-divisors function of n; •γis the Euler-Mascheroni constant (γ≈0.5772); •βis a constant in the interval (0,1 2), related to the non-trivial zeros of the Riemann zeta function. 6 2.1.4 On the Espinosa-Riemann Equivalence Theorem Finally, given the brief preceding Sections, the central result, Theorem 4, of this work is presented, which allows generating equivalences of the famous problem in an increasingly faster, compact, and most importantly beautiful way. ACorollary 2 is also derived. It begins with the following result: Proposition 4. Let T:R+→Rbe a function such that for every natural number k∈N, the inequality holds: T(k)> Hk−γ and let O:R+→Rbe the function defined (convergent in its entire domain): O(x):=X k∈N xk k!T(k). Then, the following inequality is satisfied for all x∈R+: O(x)> exlog x. Proof. Fix x>0arbitrarily. We take the definition of the function O(x): O(x):=X k∈N xk k!T(k). As we know by hypothesis that for every integer k≥1: Hk−γ < T(k). Multiplying both sides of the inequality Hk−γ < T(k)by xk k!(which is a positive term, since x>0), we obtain: (Hk−γ)xk k!<T(k)xk k!. Summing over all values of k∈N(from k= 1 to ∞), we establish the following relation for the series: X k∈N (Hk−γ)xk k!<X k∈N T(k)xk k!. By definition, the right side of this inequality is precisely O(x), from which: X k∈N (Hk−γ)xk k!<O(x). 7 Now, let’s consider the left part of this inequality. By Proposition 3, we know that for any x∈R+: exln x+1 2ln 1 + 2 x+γ < X k∈N (Hk−γ)xk k!. Combining the two obtained inequalities, we have the following chain: exln x+1 2ln 1 + 2 x+γ < X k∈N (Hk−γ)xk k!<O(x). From this chain of inequalities, we can directly infer that: exln x+1 2ln 1 + 2 x+γ < O(x). Given that x∈R+, the terms 1 2ln 1 + 2 xand γare both positive. Specifically, ln 1 + 2 x>0because 1 + 2 x>1, and the Euler-Mascheroni constant γ≈0.57721 is positive. Therefore, the sum of these two terms, 1 2ln 1 + 2 x+γ, is strictly greater than zero. This implies that: exln x < exln x+ 1 2ln 1 + 2 x+γ!. Finally, by combining this last inequality with the one we established previously: exln x < exln x+1 2ln 1 + 2 x+γ!<O(x), we conclude that: exln x < O(x). This proof is valid for all x∈R+. That said, here is the Espinosa-Riemann Equivalence Theorem: Theorem 4 (Espinosa-Riemann Equivalence Theorem).The Riemann Hypothesis is equivalent to: O(Hn) = X k∈N Hk n k!T(k)> σ(n) for all n∈N, where Hn=Pn i=1 1 iis the n-th harmonic number, σ(n) = Pd|nd is the sum-of-divisors function of nand the functions O:R+→Rand T: R+→Rsatisfy the following conditions: 8 1. Asymptotic behavior of O(x): The function satisfies the asymptotic behavior: O(x):=X k∈N xk k!T(k)=exlog x+Oex xwhen x→ ∞. 2. Condition on the inner function for T(k): For all k∈Nthe inequality holds: T(k)> Hk−γ. 3. Validation condition for small n: The inequality O(Hn)> σ(n)holds for all natural numbers nin the range 1≤n≤60. Proof. (Riemann Hypothesis ⇒Inequality): Assume the Riemann Hypothesis is valid. By Corollary 1, for all integers n>60 it holds that: σ(n)< eHnlog Hn where Hn=Pn k=1 1 kis the n-th harmonic number. By the definition of the function O(x), which is defined for all x∈R+by: O(x) = X k∈N xk k!T(k). And by the Condition on the inner function for T(k), for all k∈Nthe inequality holds: T(k)> Hk−γ. We apply Proposition 4 to O(x), from which we obtain that the following inequality is satisfied for all x∈R+: O(x)> exlog x. Evaluating at x=Hn>0, since Hn∈R+for n∈N, we obtain: O(Hn)> eHnlog Hn. Combining these inequalities, for n > 60 we have: σ(n)< eHnlog Hn<O(Hn) = X k∈N Hk n k!T(k). But by the Validation condition for small n(by hypothesis) we have that: O(Hn)> σ(n) also holds for the integers 1≤n≤60. Therefore, the inequality is valid for all n∈N. (Inequality ⇒Riemann Hypothesis): 9