Five Equivalences of the Riemann Hypothesis (The Riemann Hypothesis and Young´s Lattice [Part 5/9])

Five Equivalences of the Riemann Hypothesis José Damián Espinosa December 15, 2025 Dedication: Στην αγαπημένη μου Θεά Μαρσέλα (A mi amada Diosa Marcela) Abstract This article proposes Five Equivalences of the Riemann Hypothesis that reduce this famous problem to the veracity of the following statement; where the coefficient akis the characteristic function of each one respectively (ak:N→R+), Hn=Pn i=1 1 iis the n-th harmonic number and σ(n) = Pd|ndis the sum-of-divisors function of n: The Riemann Hypothesis is equivalent to: X k∈N Hk n k!ak> σ(n) for all n∈N. “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 no simpler.”– Albert Einstein 1 “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 “Logic will get you from A to B. Imagination will take you everywhere.”– Albert Einstein 2 “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 4 2 Content 4 2.1 Five Equivalences of the Riemann Hypothesis . . . . . . . . . . . 4 2.1.1 Espinosa’s Riemann Hypothesis Equivalence (2025) with T(k) = g ln k.......................... 4 2.1.2 Espinosa’s Riemann Hypothesis Equivalence (2025) with T(k) = ln(k+ 1) ....................... 8 2.1.3 Espinosa’s Riemann Hypothesis Equivalence (2025) with T(k)=η(k).......................... 11 2.1.4 Espinosa’s Riemann Hypothesis Equivalence (2025) with T(k)=τ(k).......................... 16 2.1.5 Espinosa’s Riemann Hypothesis Equivalence (2025) with T(k)=M(k)......................... 20 2.1.6 Analysis and comparison of all proposed Equivalences, “TheBig5” .......................... 27 2.2 On Certain Deductible Functions . . . . . . . . . . . . . . . . . . 33 References 37 3 1 Introduction This work states Five Equivalences of the Riemann Hypothesis, presented and proven in the articles, A Beautiful Equivalence of the Riemann Hypothesis [Esp25a], Two T(k)-Integral Equivalences of the Riemann Hypothesis [Esp25c], Two T(k)-Summation Equivalences of the Riemann Hypothesis [Esp25d], respectively, using the central result and consequences of the work, On a Theorem of Equivalences of the Riemann Hypothesis [Esp25b]. 2 Content This Section 2 presents the equivalences of the famous and infamous problem in Section 2.1 and a small gift, a surprise, in Section 2.2. 2.1 Five Equivalences of the Riemann Hypothesis 2.1.1 Espinosa’s Riemann Hypothesis Equivalence (2025) with T(k) = g ln k Theorem 1 (Espinosa, 2025).The Riemann Hypothesis is equivalent to: X k∈N Hk n k!g ln 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 n, •g ln k= ln k+1 k(modified natural logarithm). Proof. See in Espinosa [Esp25c] (Two T(k)-Integral Equivalences of the Riemann Hypothesis). Corollary 1 (Integral formulation).The Riemann Hypothesis is equivalent to: X k∈N Hk n k!ck> 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 n, 4 •ckis the sequence defined by: ck= 1 + Zk 11 x−1 kdx. Proof. See in Espinosa [Esp25c] (Two T(k)-Integral Equivalences of the Riemann Hypothesis). Figure 1: Bound of Espinosa’s Inequality from Theorem 1 for the function σ(n), for values of n∈Nwith n≤100. 5 Figure 2: Bound of Espinosa’s Inequality from Theorem 1 for the function σ(n), for values of n∈Nwith n≤1000. Figure 3: Bound of Espinosa’s Inequality from Theorem 1 for the function σ(n), for values of n∈Nwith n≤10000. 6 Figure 4: Coefficient akof the Bound of Espinosa’s Inequality from Theorem 1for the function σ(n), for values of k∈Nwith k≤100. Figure 5: Coefficient akof the Bound of Espinosa’s Inequality from Theorem 1for the function σ(n), for values of k∈Nwith k≤1000. 7 2.1.2 Espinosa’s Riemann Hypothesis Equivalence (2025) with T(k) = ln(k+ 1) Theorem 2 (Espinosa, 2025).The Riemann Hypothesis is equivalent to: X k∈N Hk n k!ln(k+ 1) > 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 n. Proof. See in Espinosa [Esp25c] (Two T(k)-Integral Equivalences of the Riemann Hypothesis). Corollary 2 (Integral formulation).The Riemann Hypothesis is equivalent to: X k∈N Hk n k!Z1+k 1 1 tdt > 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 n. Proof. See in Espinosa [Esp25c] (Two T(k)-Integral Equivalences of the Riemann Hypothesis). Corollary 3 (Explicit expansion).The inequality of Theorem 2 expands as: H1 n 1! Z2 1 1 tdt +H2 n 2! Z3 1 1 tdt +H3 n 3! Z4 1 1 tdt +···> σ(n). Proof. See in Espinosa [Esp25c] (Two T(k)-Integral Equivalences of the Riemann Hypothesis). 8 Figure 6: Bound of Espinosa’s Inequality from Theorem 2 for the function σ(n), for values of n∈Nwith n≤100. Figure 7: Bound of Espinosa’s Inequality from Theorem 2 for the function σ(n), for values of n∈Nwith n≤1000. 9 2.1.4 Espinosa’s Riemann Hypothesis Equivalence (2025) with T(k) = τ(k) Theorem 4 (Espinosa, 2025).The Riemann Hypothesis is equivalent to: X k∈N Hk n k!τ(k)> σ(n) for all n∈N, where: •τ(k) = Pk i=1 1 ih(i) k+1; •h(p) m=Pm j=2 1 jpdenotes the m-th generalized harmonic number of order p starting from j= 2; •Hn=Pn i=1 1 iis the n-th harmonic number; •σ(n) = Pd|ndis the sum-of-divisors function of n. Proof. See in Espinosa [Esp25d] (Two T(k)-Summation Equivalences of the Riemann Hypothesis). Corollary 8 (Espinosa, 2025).The Riemann Hypothesis is equivalent to: H1 n 1! 1 1h(1) 2+H2 n 2! 1 1h(1) 3+1 2h(2) 3+H3 n 3! 1 1h(1) 4+1 2h(2) 4+1 3h(3) 4+···> σ(n) for all n∈N, where: •h(p) m=Pm j=2 1 jpdenotes the m-th generalized harmonic number of order p starting from j= 2; •Hn=Pn i=1 1 iis the n-th harmonic number; •σ(n) = Pd|ndis the sum-of-divisors function of n. Proof. See in Espinosa [Esp25d] (Two T(k)-Summation Equivalences of the Riemann Hypothesis). Corollary 9 (Espinosa, 2025).The Riemann Hypothesis is equivalent to: X k∈N Hk n k!ak> σ(n) for all n∈N, where: •The coefficient akis defined recursively as: ak=ak−1+Pk−1 i=1 1 i(k+1)i+ 1 kh(k) k+1 for k > 1, with a1=1 2; 16 •h(p) m=Pm j=2 1 jpdenotes the m-th generalized harmonic number of order p starting from j= 2; •Hn=Pn i=1 1 iis the n-th harmonic number; •σ(n) = Pd|ndis the sum-of-divisors function of n. Proof. See in Espinosa [Esp25d] (Two T(k)-Summation Equivalences of the Riemann Hypothesis). Figure 16: Bound of Espinosa’s Inequality from Theorem 4 for the function σ(n), for values of n∈Nwith n≤100. 17 Figure 17: Bound of Espinosa’s Inequality from Theorem 4 for the function σ(n), for values of n∈Nwith n≤1000. Figure 18: Bound of Espinosa’s Inequality from Theorem 4 for the function σ(n), for values of n∈Nwith n≤10000. 18 Figure 19: Coefficient akof the Bound of Espinosa’s Inequality from Theorem 4for the function σ(n), for values of k∈Nwith k≤100. Figure 20: Coefficient akof the Bound of Espinosa’s Inequality from Theorem 4for the function σ(n), for values of k∈Nwith k≤1000. 19 2.1.5 Espinosa’s Riemann Hypothesis Equivalence (2025) with T(k) = M(k) Theorem 5 (Espinosa, 2025).The Riemann Hypothesis is equivalent to: X k∈N Hk n k!M(k)> σ(n) for all n∈N, where: •M(k) = Pk2 i=k 1 iis the function defined by the sum; •Hn=Pn i=1 1 iis the n-th harmonic number; •σ(n) = Pd|ndis the sum-of-divisors function of n. Proof. See in Espinosa [Esp25a] (A Beautiful Equivalence of the Riemann Hypothesis). Corollary 10 (Espinosa, 2025).The Riemann Hypothesis is equivalent to: X k∈N Hk n k!ak> σ(n) for all n∈N, where: •Hn=1 1+· · · +1 n; •ak=1 k+· · · +1 k2; •σ(n) = Pd|ndis the sum-of-divisors function of n. Proof. See in Espinosa [Esp25a] (A Beautiful Equivalence of the Riemann Hypothesis). Corollary 11 (Espinosa, 2025).The Riemann Hypothesis is equivalent to: X k∈N1 1+· · · +1 nk k!1 k+···+1 k2>X d|n d for all n∈N. Proof. See in Espinosa [Esp25a] (A Beautiful Equivalence of the Riemann Hypothesis). 20 Figure 21: Bound of Espinosa’s Inequality from Theorem 5 for the function σ(n), for values of n∈Nwith n≤100. Figure 22: Bound of Espinosa’s Inequality from Theorem 5 for the function σ(n), for values of n∈Nwith n≤1000. 21 Figure 23: Bound of Espinosa’s Inequality from Theorem 5 for the function σ(n), for values of n∈Nwith n≤10000. Figure 24: Coefficient akof the Bound of Espinosa’s Inequality from Theorem 5for the function σ(n), for values of k∈Nwith k≤100. 22 Figure 25: Coefficient akof the Bound of Espinosa’s Inequality from Theorem 5for the function σ(n), for values of k∈Nwith k≤1000. Figure 26: Bound of Espinosa’s Inequality from Theorem 5 for the function σ(n), for values of n∈Nwith n≤1000. 23 Figure 27: Bound of Espinosa’s Inequality from Theorem 5 for the function σ(n), for values of n∈Nwith n≤1000. Figure 28: Bound of Espinosa’s Inequality from Theorem 5 for the function σ(n), for values of n∈Nwith n≤10000. 24 Figure 29: Bound of Espinosa’s Inequality from Theorem 5 for the function σ(n), for values of n∈Nwith n≤10000. Figure 30: Bound of Espinosa’s Inequality from Theorem 5 for the function σ(n), for values of n∈Nwith n≤10000. 25 Figure 42: Bound of the Inequality with coefficient akfor each Equivalence of the Riemann Hypothesis, of the function σ(n), for values of n∈Nwith n≤10000. Figure 43: Bound of Espinosa’s Inequality from Theorem 5 for the function σ(n), for values of n∈Nwith n≤10000. 32 2.2 On Certain Deductible Functions Definition 1 (Function E(n)).The function E(n)is defined as follows: E(n) =        Hn,if nis prime Pm i=1 riHpi,if n=pr1 1pr2 2···prm mis composite 1,if n= 1 (1) where: •Hnis the n-th harmonic number, Hn=Pn k=1 1 k. •The second line applies when nis a composite number, and is decomposed into its prime factors p1, p2, . . . , pmwith their respective exponents r1, r2, . . . , rm. Definition 2 (Function F(n)).The function F(n)is defined as follows: F(n) =        h(1) n,if nis prime Pm i=1 rih(1) pi,if n=pr1 1pr2 2···prm mis composite 1,if n= 1 (2) where: •h(p) n=Pn j=2 1 jpis the n-th generalized harmonic number starting from j= 2. •The second line applies when nis a composite number, and is decomposed into its prime factors p1, p2, . . . , pmwith their respective exponents r1, r2, . . . , rm. 33 Figure 44: Function E(n)for 1≤n≤100. Figure 45: Function E(n)for 1≤n≤1000. 34 Figure 46: Function E(n)for 1≤n≤10000. Figure 47: Function F(n)for 1≤n≤100. 35 Figure 48: Function F(n)for 1≤n≤1000. Figure 49: Function F(n)for 1≤n≤10000. 36 References [Apo76] T. M. Apostol. Introduction to Analytic Number Theory. Springer, 1976. url:https://dl.icdst.org/pdfs/files1/ebc2974176a03a b93756026a97b6d370.pdf. [AS64] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Dover, 1964. url:https://archive.org/details/handbook ofmathem1964abra. [Bru81] N. G. de Bruijn. Asymptotic Methods in Analysis. Dover, 1981. url: https://store.doverpublications.com/0486642216.html. [Cor09] T. H. Cormen. Introduction to Algorithms. 3rd ed. MIT Press, 2009. url:https://mitpress.mit.edu/books/introduction-algorith ms. [Din73] R. B. Dingle. Asymptotic Expansions: Their Derivation and Interpretation. London: Academic Press, 1973. url:https://michaelbe rryphysics.wordpress.com/wp-content/uploads/2013/07/ding le.pdf. [Esp25a] J. D. Espinosa. “A Beautiful Equivalence of the Riemann Hypothesis”. In: Zenodo (2025). doi:10.5281/zenodo.17779054.url:http s://doi.org/10.5281/zenodo.17779054. [Esp25b] J. D. Espinosa. “On a Theorem of Equivalences of the Riemann Hypothesis”. In: Zenodo (2025). doi:10.5281/zenodo.17834714.url: https://doi.org/10.5281/zenodo.17834714. [Esp25c] J. D. Espinosa. “Two T(k)-Integral Equivalences of the Riemann Hypothesis”. In: Zenodo (2025). doi:10.5281/zenodo.17850259.url: https://doi.org/10.5281/zenodo.17850259. [Esp25d] J. D. Espinosa. “Two T(k)-Summation Equivalences of the Riemann Hypothesis”. In: Zenodo (2025). doi:10.5281/zenodo.17923006. url:https://doi.org/10.5281/zenodo.17923006. [Lag02] J. C. Lagarias. “An Elementary Problem Equivalent to the Riemann Hypothesis”. In: American Mathematical Monthly (2002). url:http s://dept.math.lsa.umich.edu/~lagarias/doc/elementaryrh.p df. [Rob84] G. Robin. “Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann”. In: Journal de Mathématiques Pures et Appliquées 63.2 (1984), pp. 187–213. url:https://www.sciencedire ct.com/science/article/pii/0021782484901676. [Rud64] W. Rudin. Principles of Mathematical Analysis. 3rd ed. McGrawHill, 1964. url:https://david92jackson.neocities.org/images /Principles_of_Mathematical_Analysis-Rudin.pdf. 37