Two $\mathfrak{T}(k)$-Integral Equivalences of the Riemann Hypothesis (The Riemann Hypothesis and Young´s Lattice [Part 3/9])

Two T(k)-Integral Equivalences of the Riemann Hypothesis José Damián Espinosa December 7, 2025 Dedication: Στην αγαπημένη μου Θεά Μαρσέλα (A mi amada Diosa Marcela) Abstract This article proposes Two T(k)-Integral Equivalences of the Riemann Hypothesis making direct use of the Espinosa-Riemann Equivalence Corollary. These equivalences take the function T(k)as a integral respectively, yielding two beautiful versions from this T(k)-Integral approach. The Riemann Hypothesis is equivalent to: X k∈N Hk n k! ] log 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 ] log k= log k+1 k(modified natural logarithm). The Riemann Hypothesis is equivalent to: X k∈N Hk n k!log(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. 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 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 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 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 Two T(k)-Integral Equivalences of the Riemann Hypothesis . . . 4 2.1.1 Preliminary Concepts . . . . . . . . . . . . . . . . . . . . 5 2.1.2 On Pk∈N(Hk−γ)xk k!..................... 8 2.1.3 On the Espinosa-Riemann Equivalence Corollary . . . . . 9 2.1.4 On S(x):=Pk∈N xk k!R∞ k1 ⌊t⌋−1 tdt ........... 10 2.1.5 On Υ(x):=Pk∈N xk k!g log k.................. 13 2.1.6 Espinosa’s Riemann Hypothesis Equivalence (2025) with T(k) = g log k.......................... 16 2.1.7 On ι(x):=Pk∈N xk k!log(k+ 1) ............... 20 2.1.8 Espinosa’s Riemann Hypothesis Equivalence (2025) with T(k) = log(k+ 1) ....................... 28 3 References 33 1 Introduction In this work, two equivalences of the Riemann Hypothesis are presented, resulting from the application of the Espinosa-Riemann Equivalence Corollary. This corollary, a natural consequence of the article On a Theorem of Equivalences of the Riemann Hypothesis [Esp25b], proposes a general classical form for a set of equivalences of the Riemann Hypothesis, a certain type, shown below: O(Hn) = X k∈N Hk n k!T(k)> σ(n) And it holds 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, and the functions O:R+→ Rand T:R+→Rmust satisfy three specific conditions that are discussed in detail and stated in Section 2.1.3. This established framework is used to propose two very similar expressions for each equivalence respectively. Although they derive and essentially share the same central idea as the main version proposed in the article A Beautiful Equivalence of the Riemann Hypothesis [Esp25a] (which also forms part of this general form of equivalences with its respective function T(k) = M(k) = Pk2 i=k 1 i). The novelty lies in defining the function T(k)by two integral expressions respectively, these two formulations differ subtly. Clearly from the previous result of the function O(Hn)it can be easily deduced that is T(k)the function that uniquely defines the equivalence. This specific structure, where T(k)is an integral expression, is referred to as the T(k)-Integral approach. This focus is made possible by the use of the generalized framework arising from the Espinosa-Riemann Equivalence Corollary. 2 Content 2.1 Two T(k)-Integral Equivalences of the Riemann Hypothesis In this Section 2.1, a continuous intertwining of results is worked with, until reaching Theorem 2 and 3, the two newly proposed versions of the Riemann Hypothesis, respectively. A set of ordered subsections is presented that provides a foundation as the narrative progresses, in a constructive manner, following the clear natural line that finally leads to the two eagerly awaited results mentioned. All necessary results for the first equivalence are addressed, leading up to it, and immediately afterward, the next one begins, developing all the missing elements, completely analogous to the first, until leading to it, the second. 4 2.1.1 Preliminary Concepts This Section 2.1.1 addresses key concepts that provide the foundation for all subsequent endeavors and constructions. We begin by defining the famous EulerMascheroni 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. We also resort to the classical integral definition of this constant γ, stated next: Definition 2 (Integral form of the Euler-Mascheroni constant γ).The constant γis defined by the improper integral (Apostol [Apo76, Theorem 3.2]): γ=Z∞ 11 ⌊t⌋−1 tdt where: •⌊x⌋is the floor function (integer part of x); •The integral converges to the value γ≈0.5772156649. Given this definition, we now have two forms of the constant γ, its classic limit form and its integral form. The reason for proceeding this way is immediately shown, as it allows us to define a sequence of integrals that are, in themselves, partial and/or incomplete forms of the total integral, which, by Definition 2, is γ, and also to propose a relationship between these and the constant itself. The following relationship is proposed for all k∈N: Proposition 1 (Partial integral of the integral form of the Euler-Mascheroni constant γ).For all k∈N, it holds that: Z∞ k1 ⌊t⌋−1 tdt =γ−Hk−1+ log k where: •⌊t⌋is the floor function (integer part of t); •γis the Euler-Mascheroni constant; •Hk−1=Pk−1 n=1 1 n(H0:= 0). 5 Proof. We develop the integral step by step: Z∞ k1 ⌊t⌋−1 tdt = lim N→∞ ZN k1 ⌊t⌋−1 tdt = lim N→∞   ⌊N⌋−1 X n=kZn+1 n 1 ndt −ZN k 1 tdt  = lim N→∞   ⌊N⌋−1 X n=k1 n·(n+ 1 −n)−(log N−log k)  = lim N→∞   ⌊N⌋−1 X n=k 1 n−log N+ log k  =  lim N→∞   ⌊N⌋−1 X n=1 1 n− k−1 X n=1 1 n−log N   + log k =lim N→∞ H⌊N⌋−1−log N−Hk−1+ log k =γ−Hk−1+ log k. (by definition of γ) As can be seen, this sequence moves away from γas kincreases and is exactly γwhen k= 1, making it a decreasing sequence of positive integrals that eventually converges to zero. However, to prove this intuition and, moreover, to bound this sequence, the following Proposition 2 is required, stated below: Proposition 2. For all k∈N, it holds that: 0<Z∞ k1 ⌊t⌋−1 tdt < 1 k where: •⌊t⌋is the floor function (integer part of t). Proof. For each n∈Nwith n≥k, we consider the interval [n, n + 1) where for t∈[n, n + 1), we have ⌊t⌋=n. In this interval, we have: 1 ⌊t⌋−1 t=1 n−1 t<1 n−1 n+ 1 =1 n(n+ 1), where the inequality is strict because 1 t>1 n+1 for all t∈[n, n + 1). Integrating this inequality over [n, n + 1): Zn+1 n1 n−1 tdt < Zn+1 n 1 n(n+ 1)dt =1 n(n+ 1), 6 since the integrand 1 n(n+1) is constant on the interval. Now, summing all these integrals from n=kto infinity: Z∞ k1 ⌊t⌋−1 tdt = ∞ X n=kZn+1 n1 n−1 tdt < ∞ X n=k 1 n(n+ 1). The series P∞ n=k 1 n(n+1) is telescopic, as seen by decomposing each term: 1 n(n+ 1) =1 n−1 n+ 1. Upon summation, all intermediate terms cancel (the −1 n+1 of one term cancels the +1 n+1 of the next): ∞ X n=k1 n−1 n+ 1=1 k−1 k+ 1+1 k+ 1 −1 k+ 2+· · · =1 k, since the other terms −1 k+1 ,+1 k+1 ,−1 k+2 ,+1 k+2 ,etc., successively cancel, and the limit of the remaining terms as n→ ∞ is zero. Therefore: Z∞ k1 ⌊t⌋−1 tdt < 1 k. For each t≥k, since n:=⌊t⌋satisfies n≤t<n+ 1, we have: 1 n−1 t>1 n−1 n+ 1 >0. Integrating over each interval [n, n + 1) and summing: Z∞ k1 ⌊t⌋−1 tdt = ∞ X n=kZn+1 n1 n−1 tdt > ∞ X n=k 0=0. Therefore, combining both results we have: 0<Z∞ k1 ⌊t⌋−1 tdt < 1 k. Having 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 3: Definition 3 (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. 7 •The radius of convergence R∈[0,+∞]is the supremum of the |x|for which the series converges. Next, we state the Series Comparison Test, which is of enormous utility in Theorem 1: Theorem 1 (Comparison Test for Series).Let {ak}k∈N,{bk}k∈N⊂R+ 0be sequences of non-negative real numbers such that: 0≤bk≤ak∀k∈N. If the series P∞ k=1 akconverges, then P∞ k=1 bkalso converges. Proof. See p. 60 in Rudin [Rud64, Theorem 3.25]. Similarly, the concept of Asymptotic Upper Bound for a real function is defined through Definition 4: Definition 4 (ONotation (Asymptotic Upper Bound)).Let f, g :N→R(or f, g :R→R). We say that: f(x) = O(g(x)) as x→ ∞, if there exist constants x0>0and M > 0such that: |f(x)| ≤ M|g(x)|for all x≥x0. And its fundamental properties are as follows in Proposition 3: Proposition 3 (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). Given all these preliminary concepts, this small section concludes, and we proceed with the next one. 2.1.2 On Pk∈N(Hk−γ)xk k! Two fundamental results for this work are proposed, which are stated below: Proposition 4. Let x∈R+, the following holds: X k∈N (Hk−γ)xk k!=exlog x+γ+O1 xas x→ ∞, where: 8 •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 [Esp25a] (A Beautiful Equivalence of the Riemann Hypothesis). Proposition 5. 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 [Esp25a] (A Beautiful Equivalence of the Riemann Hypothesis). Up to this point, we have the series Pk∈N(Hk−γ)xk k!, along with two sharp bounds, one upper and one lower, respectively, and an asymptotic expression. Therefore, we have everything required for it, so we proceed to the next section. 2.1.3 On the Espinosa-Riemann Equivalence Corollary In this Section 2.1.3, the result mentioned in the Introduction of this work is presented, which allows constructing equivalences of the Riemann Hypothesis! It is stated below: Corollary 1 (Espinosa-Riemann Equivalence Corollary).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 n, and the functions O:R+→Rand T: R+→Rsatisfy the following conditions: 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as x→ ∞. 2. Strict lower bound of O(k): For all x∈R+the following inequality is satisfied: O(x)> exlog x. 3. Validation condition for small n: The inequality O(Hn)> σ(n)holds for all natural numbers nin the range 1≤n≤60. 9 We consider the Corollary 2, which reduces the Strict lower bound for Υ(x) to a nice expression, as follows: Corollary 2 (Dominant growth of Υ(x)).For all x>0, the following holds: Υ(x)> exlog x. Proof. Since for x>0the following holds: 1 2log 1 + 2 x>0and γ > 0. And from the Proposition 12 we know that: Υ(x)> exlog x+1 2log 1 + 2 x+γ. Then: Υ(x)> exlog x. Thus, the function Υ(x)is now fully defined and its analytical properties, including bounds and asymptotic equivalences, have been established. We conclude this section and proceed to the next, where the central issue of this work, the famous Riemann Hypothesis. 2.1.6 Espinosa’s Riemann Hypothesis Equivalence (2025) with T(k) = g log k In this Section 2.1.6 is presented the first important result, the first proposed equivalence of the coveted Riemann Hypothesis in this article. Therefore, we proceed to state the Theorem 2: Theorem 2 (Espinosa, 2025).The Riemann Hypothesis is equivalent to: X k∈N Hk n k!g log 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 log k= log k+1 k(modified natural logarithm). Proof. By the Corollary 1, taking the functions O(x) = Υ(x)and T(x) = ] log k, and since the three required conditions for its application are met respectively, the Asymptotic behavior of O(x)condition by the Proposition 11, the Strict lower bound of O(k)by the Corollary 2, and the Validation condition for small nis verified, since the inequality O(Hn)> σ(n)holds for all natural numbers nin the range 1≤n≤60. The desired result then follows. 16 Figure 1: Bound of Espinosa’s Inequality from Theorem 2 for the function σ(n), for values of n∈Nwith n≤100. Figure 2: Bound of Espinosa’s Inequality from Theorem 2 for the function σ(n), for values of n∈Nwith n≤1000. 17 Figure 3: Bound of Espinosa’s Inequality from Theorem 2 for the function σ(n), for values of n∈Nwith n≤10000. Figure 4: Coefficient akof the Bound of Espinosa’s Inequality from Theorem 2 for the function σ(n), for values of k∈Nwith k≤100. 18 Figure 5: Coefficient akof the Bound of Espinosa’s Inequality from Theorem 2 for the function σ(n), for values of k∈Nwith k≤1000. It should be noted that it is possible to rewrite the equivalent statement from the Theorem 2 in its integral form, from which it is deduced that T(x) = ] log k is T(x)-Integral. The result is presented in the Corollary 3 as follows: Corollary 3 (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, •ckis the sequence defined by: ck= 1 + Zk 11 x−1 kdx. 19 Proof. Proof that ck=] log k. ck= 1 + Zk 11 x−1 kdx = 1 + log x−x kk 1 = 1 + (log k−1) −log 1 −1 k = log k+1 k =] log k. Therefore, since the expression is identical to the one proposed by the Theorem 2 for all n∈N, the proposed equivalence follows. The first T(k)-Integral Equivalence of the Riemann Hypothesis is thus proven. Now, a slightly modified second version is proposed, without the use of ] log k, simpler and even with a bound more adjusted. This section is therefore concluded. 2.1.7 On ι(x):=Pk∈N xk k!log(k+ 1) We begin by defining the following function analogous to the function Υ(x) through Definition 7 as follows: Definition 7 (Function ι).Let x∈R+, it is defined as: ι(x):=X k∈N xk k!log(k+ 1). Having done this, the following property is stated in Proposition 13 in the following way: Proposition 13 (Logarithmic inequality).For all x > 0, the following holds: 1 x+ 1 <log 1 + 1 x<1 x. Proof. See p. 68 in Abramowitz [AS64, Formula 4.1.33]. Next, we proceed to prove the convergence of ι(x)using Proposition 13 in Proposition 14 as follows: Proposition 14. The series ι(x)converges for all x∈R+. 20 Proof. We fix arbitrary x>0. Since we have the following equality: log(k+ 1) = log k+ log 1 + 1 k. By Proposition 13, we have the fundamental bound: 0<log(k+ 1) = log k+ log 1 + 1 k<log k+1 kfor all k∈N. Multiplying by xk k!(which is >0): 0<xk k!log(k+ 1) <xk k!log k+1 k. We define: ak:=xk k!log k+1 k, bk:=xk k!log(k+ 1). It is verified: 0≤bk≤ak∀k∈N. By the Theorem 1, and Proposition 10 since ∞ X k=1 ak= ∞ X k=1 xk k!log k+1 k= ∞ X k=1 xk k! ] log k= Υ(x) converges, then P∞ k=1 bk=ι(x)converges. Since the conclusion holds for all x>0, by arbitrariness of x, it converges for all x∈R+. Next, a vital relationship with Υ(x)is stated in Proposition 15: Proposition 15 (Exact relationship between Υ(x)and ι(x)).For all x > 0, the following holds: Υ(x)−X k∈N xk k!k(k+ 1) < ι(x)<Υ(x) where: •Υ(x):=Pk∈N xk k!log k+1 k; •ι(x):=Pk∈N xk k!log(k+ 1). Proof. We prove each inequality separately: 21 Upper Bound (ι(x)<Υ(x)): log(k+ 1) = log k+ log 1 + 1 k <log k+1 k(by Proposition 13) ⇒xk k!log(k+ 1) <xk k!log k+1 k ⇒ ∞ X k=1 xk k!log(k+ 1) < ∞ X k=1 xk k!log k+1 k ⇒ι(x)<Υ(x). Lower Bound (Υ(x)−Pxk k!k(k+1) < ι(x)): log(k+ 1) = log k+ log 1 + 1 k >log k+1 k+ 1 (by Proposition 13) = log k+1 k−1 k+1 k+ 1 = log k+1 k−1 k(k+ 1) ⇒xk k!log(k+ 1) >xk k!log k+1 k−xk k!k(k+ 1) ⇒ ∞ X k=1 xk k!log(k+ 1) >Υ(x)− ∞ X k=1 xk k!k(k+ 1) ⇒ι(x)>Υ(x)− ∞ X k=1 xk k!k(k+ 1). (Proof of ι(x)<Υ(x)) We start by analyzing the general term log(k+ 1). We decompose it using logarithm properties: log(k+ 1) = log k+ log 1 + 1 k. Now we apply the logarithmic inequality from Proposition 13, which tells us that: log 1 + 1 k<1 k. Substituting this into our previous expression: log(k+ 1) <log k+1 k. 22 We multiply both sides by xk k!(which is always positive for x>0): xk k!log(k+ 1) <xk k!log k+1 k. Finally, we sum this inequality for all k≥1: ∞ X k=1 xk k!log(k+ 1) < ∞ X k=1 xk k!log k+1 k. which directly gives us: ι(x)<Υ(x). (Proof of Υ(x)−Pxk k!k(k+1) < ι(x)) Again, we start from the decomposition: log(k+ 1) = log k+ log 1 + 1 k. This time we apply the left side of Proposition 13: log 1 + 1 k>1 k+ 1. We rewrite the term 1 k+1 as: 1 k+ 1 =1 k−1 k(k+ 1). Substituting into the original expression: log(k+ 1) >log k+1 k−1 k(k+ 1). We multiply by xk k!: xk k!log(k+ 1) >xk k!log k+1 k−xk k!k(k+ 1). Summing for all k≥1: ∞ X k=1 xk k!log(k+ 1) > ∞ X k=1 xk k!log k+1 k− ∞ X k=1 xk k!k(k+ 1). Which gives us the desired inequality: ι(x)>Υ(x)− ∞ X k=1 xk k!k(k+ 1). 23 We now proceed to improve the inequality from Proposition 2 as follows in Proposition 16: Proposition 16 (Improvement of Proposition 2).For all k∈N: 1 k(k+ 1) <Z∞ k1 ⌊t⌋−1 tdt < 1 k. Proof. For each n≥k, in [n, n + 1) where ⌊t⌋=n: Zn+1 n1 n−1 tdt =1 n−log 1 + 1 n= ∞ X m=2 (−1)m mnm. The terms are grouped into positive pairs: 1 2n2−1 3n3+1 4n4−1 5n5+· · · >0 since n≥1, and for each p≥1: 1 (2p)n2p−1 (2p+ 1)n2p+1 ≥0.(since (2p+ 1)n>2p) We bound for n≥2, using the first two terms: 1 n−log 1 + 1 n=1 2n2−1 3n3+1 4n4−1 5n5+· · · >1 2n2−1 3n3≥1 3n2 since 1 2n2−1 3n3≥1 3n2for n≥2. The infinite integral is the sum of these terms: Z∞ k1 ⌊t⌋−1 tdt = ∞ X n=kZn+1 n1 n−1 tdt = ∞ X n=k 1 n−log 1 + 1 n!. Summing from n=k≥2: ∞ X n=k 1 n−log 1 + 1 n!>1 3 ∞ X n=k 1 n2>1 3 ∞ X n=k 1 n(n+ 1) >1 3k. For k≥2:1 3k≥1 k(k+ 1). Therefore we have: Z∞ k1 ⌊t⌋−1 tdt = ∞ X n=k 1 n−log 1 + 1 n!>1 k(k+ 1) when k≥2. The case k= 1 is also satisfied, since R∞ 11 ⌊t⌋−1 tdt =γ > 1 2=1 1(2) .Therefore we have for each k≥1: Z∞ k1 ⌊t⌋−1 tdt > 1 k(k+ 1). 24 But from Proposition 2, for all k∈N: Z∞ k1 ⌊t⌋−1 tdt < 1 k. Therefore, for all k∈Nwe have: 1 k(k+ 1) <Z∞ k1 ⌊t⌋−1 tdt < 1 k. Which gives us the desired inequality. Now we proceed to state a corollary and a proposition, necessary for what is proposed in the present Section 2.1.8; The corollary gives a fine lower bound for ι(x)analogous to Corollary 2 of Υ(x)respectively: Corollary 4 (Dominant growth for ι(x)).For all x>0, the following holds: ι(x)> exlog x. Proof. For each k≥1, we have by Proposition 16: Z∞ k1 ⌊t⌋−1 tdt > 1 k(k+ 1). Since x>0and k!>0, each term xk k!is positive. Multiplying: xk k!Z∞ k1 ⌊t⌋−1 tdt > xk k! 1 k(k+ 1). Summing over all k≥1: S(x) = ∞ X k=1 xk k!Z∞ k1 ⌊t⌋−1 tdt > ∞ X k=1 xk k! 1 k(k+ 1). For the second term, by Proposition 5: ∞ X k=1 (Hk−γ)xk k!> exlog x+1 2log 1 + 2 x+γ. Combining both results: Υ(x)=S(x) + ∞ X k=1 (Hk−γ)xk k!> ∞ X k=1 xk k!k(k+ 1) +exlog x+1 2log 1 + 2 x+γ. Subtracting P∞ k=1 xk k!k(k+1) from both sides of the inequality we obtain: Υ(x)− ∞ X k=1 xk k!k(k+ 1) > exlog x+1 2log 1 + 2 x+γ. 25 Figure 9: Coefficient akof the Bound of Espinosa’s Inequality from Theorem 3 for the function σ(n), for values of k∈Nwith k≤100. Figure 10: Coefficient akof the Bound of Espinosa’s Inequality from Theorem 3 for the function σ(n), for values of k∈Nwith k≤1000. 32 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. 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