The Elementary Proof of the Riemann's Hypothesis

This research paper aims to explicate the complex issue of the Riemann's Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to demonstrate the maximal prime gaps bound. Maximal prime gaps bound constitutes a comprehensive improvement over the Bertrand's result, and becomes one of the key elements of the theory. Subsequently, implementing the theory of the primorial function and its error bounds, an improved version of the Gauss' offset logarithmic integral is developed. This integral serves as a Supremum bound of the prime counting function Pi(n). Due to its very high precision, it permits to verify the relationship between the prime counting function Pi(n) and the offset logarithmic integral of Carl Gauss. The collective mathematical theory, via the Niels F. Helge von Koch equation, enables to prove the RIemann's Hypothesis conclusively.


Definitions section
Within the scope of the paper, prime gap of the size g ∈ N | g ≥ 2 is defined as an interval between two primes (p i , p i+1 ], containing (g − 1) composite integers. Maximal prime gap of the size g, is a gap strictly exceeding in size any preceding gap. In this document, all computations pertaining to the logarithmic integral, were carried out using the Gauss' offset logarithmic integral : where ζ ′ is the derivative of the Riemann zeta function. where K is the Khinchin's constant.

Preliminaries.
Bertrand's Conjecture is a well known mathematical theorem concerning the size of the prime gaps. The first elementary proof of the Bertrands Conjecture regarding the existence of at least one prime within the interval from n to 2n was due to Srinivasa Ramanujan, who in 1919 presented his elegant proof. Paul Erdös at the age of 19 improved Ramanujan's proof in 1932. In his proof of the Bertrand's conjecture Paul Erdös utilized the largest binomial coefficient of the binomial expansion 2 2n : The problem of existence of at least one prime within the interval from n to n + c = t is substantially more difficult than the Bertrand's Conjecture. The issue pertains to the considerably shorter interval length of the function G (n) , as compared to the interval of length n, pertinent to the research that both Srinivasa Ramanujan and Paul Erdös worked on.
One of the major step-stones of this paper is the comprehensively improved bound on the maximal prime gaps. This goal is achieved by an implementation of a binomial expansion coefficient pertinent to the function G (n) . Now, for all n ∈ N | n ≥ 5, we make the following definitions: Consequently, implementing the lower/upper bounds on the logarithm of n! for the bounds on log M (t) , results in bounds of the form: (2.5) log Keeping the values of c, n and t constant and letting the variable k to increase unboundedly, results in an unbounded monotonically decreasing function. When implementing the lower/upper bounds on the logarithm of n! for the Supremum/Infimum bounds on log M (t) , the variable k appears only with values k = {0, 1} respectively. The combined effect of the difference of the logarithms of factorial terms in log M (t) and the decreasing property of the function 2.5, imposes a reciprocal interchange of the bounds 2.1, when implementing them for the bounds on log M (t) .

Lemma 2.6 (log M (t) Supremum Bound).
The Supremum Bound on the logarithm of the binomial coefficient M (t) is given by:
Consequently, from Lemma 2.7 and 2.6 we have: (2.14) log Inequality 2.14 presents very well streamlined Supremum/Infimum bounds on the log M (t) .
3. Maximal prime gaps Figure 1. The left drawing shows the graphs of the lower (blue) and upper (red) bounds vs log M (t) (black). The right drawing shows the graph of G (n) (red) and the actual maximal gaps (black) with respect to ξ as given by the Definition 5.4. The graph has been produced on the basis of data obtained from C. Caldwell as well as from T. Nicely tables of maximal prime gaps.
From the Prime Number Theorem we have that an average gap between consecutive primes is given by log n for any n ∈ N. There exist however prime gaps much shorter -containing only a single composite number, and gaps which are much longer than average -the maximal prime gaps. In 1929 R. Backlund [1] published a paper in which he proved the lower bound on the maximal prime gaps: p (n+1) − p (n) > (2 − ϵ) log p (n) for any ϵ > 0 This was the first major result in this area. It had been improved upon in 1935 by Paul Erdös [14] who proved that: c(log p (n) ) log(log p (n) ) (log(log(log p (n) ))) 2 However, it was the pioneering work of H. Cramér [12] using sophisticated probabilistic techniques, who attempted to establish the upper bound on the maximal prime gaps: p (n+1) − p (n) ≤ (log p (n) ) 2 We begin with a preliminary derivation. Since the integers from 1 to n contain ⌊ n p ⌋ multiples of the prime number p, ⌊ n p 2 ⌋ multiples of p 2 etc. Thus it follows that: In accordance with the definitions 2.1 of G (n) , 2.2 of t and 2.3 of M (t) we obtain: Where p is as usual a prime number. Let's define: The case when there does not exist any prime factor p of M (t) within the interval from n to (n + G (n) ) = t for any n ∈ N | n ≥ 8, imposes an upper limit on all prime factors p of M (t) . Consequently in this particular case, every prime factor p must be less than or equal to s = ⌊ t 2 ⌋ .

Proof.
Let p be a prime factor of M (t) so that K p ≥ 1 and suppose that every prime factor p ≤ n. If s < p ≤ n then, p < (n + G (n) ) < 2p and and so K p = 0. Therefore p ≤ s for every prime factor p of M (t) , for any n ∈ N | n ≥ 8.

Maximal prime gaps standard measure.
The binomial coefficient M (t) : The bounds on the logarithm of M (t) are given by Lemma 2.6 and 2.7: • The function Gs (n) due to the implementation of the Floor function increases stepwise. The sudden increase in value of the function Gs (n) is mirrored by an analogous, simultaneous increase in both, implemented bounds on the function log M (t) as well as the function log M (t) itself.
Theorem 3.6 (Maximal Prime Gaps Bound and Infimum for primes). For any n ∈ N | n ≥ 8 there exists at least one p ∈ N | n < p ≤ n + G (n) = t; where p is as usual a prime number and the maximal prime gaps upper bound G (n) is given by:

Proof.
Suppose that there is no prime within the interval from n to t. Then in accordance with the hypothesis, by Lemma 3.2 we have that, every prime factor p of M (t) must be less than or equal to s = ⌊ t 2 ⌋ . Invoking Definitions 3.3, 3.4 and 3.5, Lemma 2.6, 2.7 and the inequality 3.1 we obtain for all n ∈ N | n ≥ 8: In accordance with the hypothesis therefore, it must be true that: Now, we apply the Cauchy's Root Test for n ≥ 43: At n = 43 the Cauchy's Root Test attains ≈ 1.17851 and tends asymptotically to 1, decreasing strictly from above. Thus, by the definition of the Cauchy's Root Test, the series formed from the terms of the difference LB (t) − UB (ts) , diverges as c increases unboundedly. Hence in accordance with the hypothesis, inequality 3.5 diverges to −∞ as n increases unboundedly. However, at n = 43 the difference 3.5 attains ∼ 9.45885151 and diverges as n increases unboundedly. Thus, we have a contradiction to the initial hypothesis. This implies that for all n ∈ N | n ≥ 43: Necessarily therefore, there must be at least one prime within the interval c for all n ∈ N | n ≥ 43. Table 1 lists all values of n s.t. 8 ≤ n ≤ 47. Evidently, every possible sub-interval contains at least one prime number. Thus we deduce that Theorem 3.6 holds in this range as well. Consequently Theorem 3.6 holds as stated for all n ∈ N | n ≥ 8, thus completing the proof. From now on, we may relax the function G (n) , by dropping the floor function.
A stronger version of the Maximal Prime Gaps bound can be proven by analogous method. However, in most instances the bound provided by Theorem 3.6 is sufficient. For the sake of completeness, Theorem 3.7 specifies the sharper, enhanced bound:

Theorem 3.7 (Sharper Maximal Prime Gaps Supremum Bound).
For any n ∈ N | n ≥ 11 there exists at least one p ∈ N | n < p ≤ n+Gs (n) = t; where p is as usual a prime number and the maximal prime gaps standard measure Gs (n) is given by: For a proof please consult Feliksiak [18].

Proof.
By Theorem 3.6 we have that there exist at least one prime p ∈ N | n < p ≤ t. Since, G (n) = 5 (log 10 n) 2 < ((log 10) (log 10 n)) 2 ∀n ∈ N | n ≥ 8 Therefore the Cramér's Maximal Gaps Conjecture follows ipso facto. There exist at least one prime p ∈ N | n 2 < p ≤ (n + 1) 2 ∀n ∈ N | n ≥ 2 Proof. Suppose that Theorem 3.9 is false for some n ∈ N| n > 10. This implies that By Theorem 3.6 we have that π n 2 < π (n 2 +G (n 2 ) ) ∀n ∈ N | n ≥ 8 Therefore, in accordance with the hypothesis it must be true that: Thus, However, for any n ∈ N | n > 10, the limit of 3.9 by the L'Hôpital's rule is: lim n→∞ 2n + 1 5 (log 10 n 2 ) 2 = lim n→∞ n(log 10) 2 20 → ∞ Hence the ratio 3.9 increases unboundedly as n tends to infinity. At n = 10, the value of the inequality 3.9 equals 1.05. It implies that: Hence we have a contradiction to the initial hypothesis. Consequently, Theorem 3.9 is satisfied for all n ∈ N | n ≥ 10. For all n ∈ N | 2 ≤ n < 10 a simple computer verification shows that Theorem 3.9 holds in this range as well, thus concluding the proof.

Upper Bound on the logarithm of the primorial function.
The natural logarithm of the primorial function is a key element of the definition of the tailored logarithmic integral. It paves the way for the estimation of the prime counting function π (n) with unparalleled accuracy. First, we define the primorial function for all k ∈ N: The natural logarithm of the primorial function is strictly less than the respective prime number p ∈ N: In particular the natural logarithm of the primorial function is asymptotic (from below) to the respective prime number: For the purpose of the proof we may assume that the twin primes continue indefinitely, the proof validity will not be affected by this. This issue will be expounded on in the Remark 4.2 below.

Proof.
From the inequality 4.1 we have: Since prime numbers continue indefinitely, both p (n) ♯ and exp ( p (n) ) , are monotonically increasing divergent sequences of positive real numbers for all n ∈ N | n ≥ 2 Suppose that Lemma 4.3 is false, in accordance with the hypothesis it implies that: However, at p n = 13 the difference 4.4 attains ∼ 412383.39201 and further diverges exponentially. Therefore, we apply the d'Alemberts Ratio Test. Define a sequence for all prime numbers p (n−1) , p (n) ∈ N| p (n) ≥ 13: The sequence a (n) given by the Definition 4.4, has the least value at the twin primes as the difference p (n) −p (n−1) = 2. Consequently, it is therefore both necessary and sufficient, to consider the sequence a (n) at the twin prime numbers only, with p (n) = 6i + 7 | i ∈ N, i ≥ 1.
At the twin primes: Further, Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 Thus, at the twin primes the sequence a (n) equals: (4.7) The bracketed expression on the RHS, at the twin primes approaches the limit: at the twin primes therefore, the sequence a (n) must clearly approach the limit: → exp (2) By the d'Alemberts Ratio Test, the series formed from the terms of the difference exp ( p (n) ) − p (n) ♯, diverges as p (n) increases unboundedly. Thus, it logically follows that: Necessarily therefore, we have a contradiction to the initial hypothesis. Since at the twin primes the sequence exp ( p (n) ) − p (n) ♯ approaches: Rearranging the above, we obtain that at the twin primes the primorial approaches: This in turn implies that a strict inequality holds: Since increasing the gap between the consecutive primes has the effect of exponentially increasing the value that the sequence a (n) attains, therefore this result holds for all p (n) ∈ N| p (n) ≥ 13. By taking the logarithms, we obtain: Thus, Lemma 4.3 holds for all p (n) ∈ N| p (n) ≥ 13. Direct computation verifies that Lemma 4.3 holds for all p (n) ∈ N| 2 ≤ p (n) ≤ 13. Therefore, Lemma 4.3 holds as stated: Consequently, this implies that the sequence of the natural logarithm of the primorial function is asymptotic from below: Concluding the proof of Lemma 4.3.
Lemma 4.3 also implies that: By the PNT, (Ruiz, 1997;Finch, 2003), and Lemma 4.3 we obtain therefore: Remark 4.2. The sequence: as it has been demonstrated for the twin primes example in the proof of Lemma 4.3; for primes such that ( for some given particular d, the sequence a (n) at the respective prime pairs, converges to the limit: The approximation improves rapidly as p (n) increases. This is the reason why the validity of the twin primes conjecture is not essential.

Lemma 4.5 (Lower Estimation Error Bound On
The Difference p n − log p n ♯).
The error of estimation of the primorial function by the use of the value of p (n) imposes the following lower bound: where γ ≈ 0.57721566490153286060651209 is the Euler-Mascheroni constant.

Proof.
Both exp ( p (n) ) and p (n) ♯ as well as exp (LB p (n) ) are monotone, divergent sequences of positive real numbers for all p (n) ∈ N | p (n) ≥ 2. Suppose that Lemma 4.5 is false. From inequality 4.21 therefore, in accordance with the hypothesis we derive: However, at p n = 13 the difference 4.22 attains ∼ 328977.240182 and further diverges exponentially. Therefore we apply the d'Alemberts Ratio Test. Define a sequence for all prime numbers p (n−1) , p (n) ∈ N| p (n) ≥ 13: Remark 4.3. The terms of the sequence a (n) given by the Definition 4.6 have the least value at the twin primes since the difference p (n) − p (n−1) = 2. Consequently, it is both necessary and sufficient, to consider the sequence 4.6 at the twin primes only, with p (n) = 6i + 7 | i ∈ N, i ≥ 1.
At the twin primes: Further: Thus, at the twin primes the sequence a (n) equals: The bracketed expression on the RHS, at the twin primes approaches the limit: at the twin primes therefore, the sequence a (n) must clearly approach the limit: → exp (2) By d'Alemberts Ratio Test, the series formed from the terms of the difference exp ( p (n) ) − p (n) ♯ exp (LB p (n) ) diverges, as p (n) increases unboundedly. Thus, it logically follows that: Necessarily therefore, we have a contradiction to the initial hypothesis. Since at the twin primes the sequence exp ( Rearranging the above we obtain that, at the twin primes the primorial approaches: This in turn implies that a strict inequality holds: Since increasing the gap between the consecutive primes has the effect of exponentially increasing the value that the sequence a (n) attains, therefore this result holds for all p (n) ∈ N| p (n) ≥ 13. By taking the logarithms, we obtain: Thus, Lemma 4.5 holds for all p (n) ∈ N| p (n) ≥ 13. Direct computation verifies that Lemma 4.5 holds for all p (n) ∈ N| 2 ≤ p (n) ≤ 13. Therefore, Lemma 4.5 holds as stated, concluding the proof.
Both exp ( p (n) ) and p (n) ♯ as well as exp (UB p (n) ) are monotone, divergent sequences of positive real numbers for all p (n) ∈ N | p (n) ≥ 2. Suppose that Lemma 4.7 is false. From inequality 4.33, in accordance with the hypothesis we derive: However, at p n = 13 the difference 4.34 attains ∼ 4.02297598 × 10 7 and further diverges exponentially. Therefore we apply the d'Alemberts Ratio Test. Define a sequence for all prime numbers p (n−1) , p (n) ∈ N| p (n) ≥ 13: Remark 4.4. The terms of the sequence a (n) given by the Definition 4.8 have the least value at the twin primes since the difference p (n) − p (n−1) = 2. Consequently, it is both necessary and sufficient, to consider the sequence 4.8 at the twin primes only, with p (n) = 6i At the twin primes: (2) Further, Thus, at the twin primes the sequence a (n) equals: The bracketed expression on the RHS, at the twin primes approaches the limit: at the twin primes therefore, the sequence a (n) must clearly approach the limit: By the d'Alemberts Ratio Test, the series formed from the terms of the difference diverges, as p (n) increases unboundedly. Thus, it logically follows that: Necessarily therefore, we have a contradiction to the initial hypothesis. Since at the twin primes the sequence ( Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 Rearranging the above we obtain, that at the twin primes the primorial approaches: This in turn implies that a strict inequality holds: Since increasing the gap between the consecutive primes has the effect of exponentially increasing the value that the sequence a (n) attains, therefore this result holds for all p (n) ∈ N| p (n) ≥ 13. By taking the logarithms, we obtain: Thus, Lemma 4.7 holds for all p (n) ∈ N| p (n) ≥ 13. Direct computation verifies that Lemma 4.7 holds for all p (n) ∈ N| 2 ≤ p (n) ≤ 13. Therefore, Lemma 4.7 holds as stated, concluding the proof.
where p is the prime p ∈ N | p ≤ n and γ is the Euler-Mascheroni constant γ ≈ 0.57721566490153286060651209.

Theorem 5.3 (Supremum and Infimum Bounds on π (n) ).
Let π n denote the prime counting function. Then, the Supremum and Infimum bounds, which constitute bounds on π n are given by: Remark 5.1. The proof is a multistep regimen and will be carried over several of the following theorems/lemmas. The proof pertains to the Supremum Bound only, an analogous process needs to be carried out in the case of the Infimum Bound, which is left at the discretion of the reader.

Preliminary theory.
Tailored logarithmic integral Supremum/Infimum bounds, as given by Theorem 5.3, present a significantly improved accuracy of estimation of the function π (n) . Lemma 4.3 states: Remark 5.2. The classical offset logarithmic integral Li (n) of C.F. Gauss, is an improvement of the estimate of the number of primes given by n/log n, up to some n ∈ N. Therefore, since the left side of the inequality 5.3 increases only at the primes as π n does, it constitutes an improvement in π (n) estimation. Numerical comparison of the performance of the Carl F. Gauss offset Li (n) vs the T Li (n) is given in Table 2, and graphically presented in Fig. 4. The graph of the tailored integral is below that of π (n) for all n ∈ N | n < 43, please refer to Fig. 6a. Since the primorial function increases only at the primes, necessarily therefore, the estimation error of the tailored integral increases at the primes only. Hence, if the relation T Li (n) ≥ π (n) holds at the primes, it therefore holds at every other point. This contrasts strongly with the Gauss' logarithmic integral Li (n) in which, the estimation error term increases over the intervals between the primes and decreases at the primes. As a result, it produces large estimation error oscillations. On the other hand T Li (n) , accurately duplicates the pattern of the curve of π (n) , with minimal error increase.
Due to the fact that T Li (n) increases stepwise at the primes, the analysis of the step size and its limit as n approaches infinity forms the core of the proof of the Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 Figure 5. The figures drawn at every n ∈ N in the range, show the graphs of Li (n) (grey), π (n) (black) and T Li (n) (red).
Remark 5.3. Both the tailored integral T Li (n) and π n are weakly monotone functions, increasing unboundedly, hence producing a positive sequence of numbers which diverges to infinity. The initial estimates of the T Li (n) step size indicate that the step sequence quickly approaches the value of 1 from above. Table 3 presents some of the values that the step sequence takes at the powers of 10.
It is obvious that the numerical value attained by the step sequence at various points fluctuates as well, as a consequence of the size of the gap between the two consecutive primes (as well as the distance to the preceding prime pair). The effect however, of the gap interval length rapidly decreases as p i increases, because the prime gaps are bounded above (Theorems 3.6 and 3.7). Lemma 5.9 (Stepwise Convergence Of The Error of Estimation of the T Li (n) ). The step sequence of the tailored logarithmic integral T Li (n) is Cauchy and converges asymptotically from above to the limit: Furthermore, the difference of the step integral T Li (n) and its approximation has the following bounds: with θ 1 and θ 2 given by the Definitions 5.5 and 5.6 respectively.

Proof.
By the Prime Number Theorem we may estimate the integral T Li (n) step sequence for any prime number p ∈ N | p ≥ 3 : Thus by the PNT we have, The logarithm of the primorial function is clearly a monotone function increasing unboundedly, hence, producing a sequence of positive real numbers which diverges to infinity. From Lemma 4.3 we have that log p (i+1) ♯ is asymptotic from below to p (i+1) , as well as: Hence, for a prime number p ∈ N, This implies that the estimating sequence converges asymptotically from above to the limit: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 Therefore it is Cauchy. The step integral T Li (n) at p 6 = 13 attains ∼ 1.13056 and the step sequence values decrease, asymptotically approaching 1 as p n increases unboundedly. Please, also refer to the Table 3. Consequently, Thus the step integral T Li (n) is Cauchy as well. Both LDB p (i+1) and U DB p (i+1) are clearly strictly monotone decreasing Cauchy sequences. Suppose that the following assertion is false: This implies that: However, at p (n) = 13 the inequality 5.11 attains ∼ 48.6109 and diverges as p (n) increases unboundedly with the rate of divergence ∝ k p (n) s.t. k ∼ 3 for larger primes p (n) . Consequently, we have a contradiction to the hypothesis. Inequality 5.10 therefore, is valid for all p n ∈ N | p n ≥ 13.
Suppose now, that the following inequality is false: This implies that: However, at p (n) = 13 the inequality 5.13 attains ∼ 3.38914 and diverges as p (n) increases unboundedly with the rate of divergence ∝ k p (n) s.t. k ∼ 1 for larger primes p (n) . Consequently, we have a contradiction to the hypothesis. Inequality 5.12 therefore, is valid for all p n ∈ N | p n ≥ 13. Necessarily this implies that the Inequality 5.5 holds as stated. This demonstrates therefore, that since U DB p (i+1) is strictly monotone decreasing Cauchy sequence with a limit L = 0: Thus, from above we have that the estimating sequence 5.8 converges asymptotically from above to its limit L = 1. Since the step integral at p (i+1) = 11 attains ∼ 1.2171 necessarily therefore the step integral tends asymptotically from above: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 This implies, that the sequence of the step estimation errors asymptotically converges from above (also refer to Table 3): Thus concluding the proof of Lemma 5.9. Remark 5.4. The integral part of the step size clearly accounts for the prime number found. Comparing each fractional part of the step (please refer to Table 3) at p i with the corresponding term of the harmonic series ( 1 pi ), it becomes obvious that it is greater than the term of the series. Since the error of estimation is the sum of π (n) of such individual terms, comparing its sum with the divergent sum of reciprocals of successive prime numbers leads to a conjecture, that the sum of the estimation error terms diverges as p i tends to infinity.

Corollary 5.10 (Infimum and Supremum
Step Sequence Estimation Error Bounds). The step sequence error of estimation of the prime counting function π (n) by the application of the tailored logarithmic integral T Li (n) ∀p (i) ∈ N | p (i) ≥ 13, is bounded below/above by: where p (i) and p (i+1) are associated with lower/upper limits of integration and θ 1 , θ 2 are given by the Definitions 5.5 and 5.6 respectively.
The Infimum and Supremum error bounds ISE (p(n)) and SSE (p(n)) for the tailored integral step estimation error are computationally very demanding. Therefore, Theorems: 5.11 and 5.12 that follow, establish simpler bounds.

Theorem 5.11 (The Step Sequence Estimation Error Lower Bound).
The estimation error of the tailored logarithmic integral T Li (n) at every step exceeds the value of the inverse of the pertinent prime number hence, it is bounded below by 1/p ∀p ∈ N | p i ≥ 13: where p (i) and p (i+1) are associated with the lower/upper limit of integration θ 1 and θ 2 respectively.

Proof.
By Lemma 5.9 the sequence SER (p(i+1)) is Cauchy and it converges from above to the limit L = 0. The sequence of the reciprocals of prime numbers is clearly Cauchy and converges to the limit L = 0. By Lemma 5.9 we have that: Consequently Theorem 5.11 is valid if and only if: as well as: Consequently, from the above we obtain: Bearing in mind that for all positive a, b ∈ R | a > b: Thus, by Lemma 4.3 we have: Hence, Suppose that the Theorem 5.11 is false. Then it must be true that the numerator of equation 5.21 is less than zero. From inequality 5.26 we see that without loss of generality, upon substitution into the numerator of the inequality 5.21, we can drop the common terms obtaining: However at p (i+1) = 37 the difference 5.27 attains ∼ 0.20084385349345676 and diverges. Hence we have a contradiction to our hypothesis which implies that the inequality is true: Consequently this implies that Theorem 5.11 is satisfied for all p i ∈ N | p i ≥ 37, a simple computer calculation verifies that this inequality also holds within the interval 13 ≤ p i ≤ 37. This necessarily means that Theorem 5.11 is satisfied for all p i ∈ N | p i ≥ 13, thus completing the proof.

Theorem 5.12 (The Step Sequence Estimation Error Upper Bound).
The inverse of a root of the pertinent prime number at every step exceeds the value of the estimation error of the tailored logarithmic integral T Li (n) step sequence ∀p i ∈ N | p i ≥ 13: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 where p (i) and p (i+1) are associated with the lower/upper limit of integration θ 1 and θ 2 respectively.
Proof. By Lemma 5.9 the sequence SER (p(i+1)) is Cauchy and it converges from above to the limit L = 0. The sequence of the reciprocals of the root of prime numbers is clearly Cauchy and converges to the limit L = 0. By Lemma 5.9 we have that: Consequently Theorem 5.12 is valid if and only if: From Lemma 4.3 we have that log p (i+1) ♯ is asymptotic (from below): as well as: From Lemma 4.5 we have for all p (i+1) ∈ N | p (i+1) ≥ 2: Consequently, from the above we obtain: Bearing in mind that for all positive a, b ∈ R | a > b: Thus, by Lemma 4.3 we have: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 Hence, Suppose that the Theorem 5.12 is false. Then it must be true that the numerator of equation 5.32 is greater than zero. From inequality 5.37 we see that without loss of generality, upon substitution into the numerator of the inequality 5.32, we can drop the common terms obtaining: However at p (i+1) = 197 the difference 5.38 attains ∼ −1.20860443 and diverges.
Hence we have a contradiction to our hypothesis which implies that the inequality is true: Consequently this implies that Theorem 5.12 is satisfied for all p i ∈ N | p i ≥ 197, a simple computer calculation verifies that this inequality also holds within the interval 13 ≤ p i ≤ 197. This necessarily means that Theorem 5.12 is satisfied for all p i ∈ N | p i ≥ 13, thus completing the proof.
Hence, by Theorems 5.11 and 5.12, for the largest prime number p (i+1) that satisfies the condition p (i+1) ≤ n ∈ N, we have: Remark 5.5. We need to re-define the lower/upper limits of integration to conform with the summation limits. The computation of the sum of step errors of the integral T Li n begins at p 2 = 3, irrespective of the fact that the computation of the sums pertinent to the bounds (Infimum, Supremum, Lower and Upper) begins first at p 15 = 47. The error arising in the estimation of the prime counting function π (n) by the application of the tailored logarithmic integral T Li (n) , diverges to infinity: Where the limits of integration θ 1 and θ 2 are given by the Definitions 5.13 and 5.14 respectively. Besides, the prime number p (n) is defined as being the biggest prime p ≤ n. Furthermore, Proof. By Lemma 5.9 for all p i ∈ N | p i ≥ 13 the relation holds at every step: Because the sum of reciprocals of successive prime numbers diverges, consequently, the sum: must necessarily diverge, by comparison with the divergent sum of reciprocals of successive prime numbers. The complete estimation error of the tailored integral is given by: Consequently therefore, by the divergence of the sum 5.44, the estimation error of the tailored integral 5.45 must necessarily diverge: In fact the sum of reciprocals of successive prime numbers and 5.45 intersect. Direct calculation at n = 983 shows that the difference: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 Because by Lemma 5.9 the inequality 5.43 holds for all p i ∈ N | p i ≥ 13, this implies that: ( Direct computation verifies that the estimation error of T Li (n) : At n = 43 the estimation error of T Li (n) attains the value of 0.002993180461560385. Therefore, by the divergence of the tailored integral estimation error, it must be true that: This concludes the proof of Theorem 5.15.

Supremum and Infimum estimation error bounds on T Li
Remark 5.6.
The estimation error of the T Li (n) increases stepwise at the primes. Since however the magnitude of the increase is small, as a result the curve presents itself as a rising virtually smooth slope. It absolutely lacks the large amplitude variation, which is the intrinsic characteristic of the Gauss' offset Li (n) . Please refer to Fig. 4. The Infimum and Supremum estimation error bounds, in conjunction with the tailored logarithmic integral, give us both the most accurate estimate of π (n) and the best estimation error bounds. The drawback is, that the formulae are computationally quite demanding.

Theorem 5.16 (Infimum Estimation Error Bound).
The error of estimation of the prime counting function π (n) by the application of the tailored logarithmic integral T Li (n) , for all n ∈ N | n ≥ 47 is bounded below by a divergent sum: Further, the limits of integration θ 1 , θ 2 implement Definition 5.13 and 5.14 respectively, while p (n) is the greatest prime p ∈ N | p ≤ n.

Proof.
Let's consider both LB (n) and IN F (n) stepwise first. Evidently, both of them are Cauchy, convergent to zero sequences, while the lower bound function LB (n) is strictly monotonic. Further, by Theorem 5.11 we have that: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 The relation holds stepwise at every step for all p (n) ∈ N | p (n) ≥ 13. Adding simultaneously the terms of both sequences and forming two respective sums, does not invalidate the relation. Considering in turn stepwise the functions IN F (n) and the step sequence T Li (n) , evidently both are Cauchy, convergent to zero sequences. Further by Lemma 5.9 and Corollary 5.10 we have that at every step: The inclusion of the additional, first step term: has clearly the effect of shifting the curve of the ( T Li (n) − π n ) significantly down, thereby upsetting the stepwise inter-relationships. However, by Theorem 5.15 we have that: with θ (n) given by the Definition 5.1. Thus, at p n = 43 the tailored integral concludes the stage of recovery instigated by the addition of the initial term, vide equation 5.48 above. The difference:

}  
at p 15 = 47 attains ∼ −0.0200272 and diverges decisively. Consequently, by Theorem 5.11 it must be true that the relation 5.50 holds for all p (n) ∈ N | p (n) ≥ 47. The difference: at p 15 = 47 the difference attains ∼ −0.0125411 and further diverges. Since the relation 5.47 holds stepwise for all p (n) ∈ N | p (n) ≥ 13, therefore, necessarily it must be true that the relation 5.51 holds for all p (n) ∈ N | p (n) ≥ 47. Consequently, for all p (n) ∈ N | p (n) ≥ 47 we obtain: This concludes the proof of Theorem 5.16.

Remark 5.7.
Inclusion of the additional, initial term: to complete the domain of integration, when summing the step terms of the T Li n up to some predetermined n ∈ N, has a drawback. It upsets the established stepwise balance with all its bounds. To resolve the issue decisively with the bounds, we have to drop 14 of the initial terms of the sum, for each bound. This way, the balance in their inter-relationships is restored, as ( T Li (n) − π (n) ) > 0 ∀n ∈ N | n ≥ 43.

Theorem 5.17 (Supremum Estimation Error Bound).
The error of estimation of the prime counting function π (n) by the application of the tailored logarithmic integral T Li (n) , for all n ∈ N | n ≥ 47 is bounded above by a divergent sum: where a = π 2 , while p (n) is the biggest prime number p ≤ n.

Proof.
Let's consider both UB (n) and SU PR (n) stepwise first. Evidently, both of them are Cauchy, convergent to zero sequences, while the upper bound function UB (n) is strictly monotonic. Further, by Theorem 5.12 we have that: the relation holds stepwise at every step for all p (n) ∈ N | p (n) ≥ 13. Adding simultaneously the terms of both sequences and forming two respective sums, does not invalidate the relation. Considering in turn stepwise the functions IN F (n) and the step sequence T Li (n) , evidently both are Cauchy, convergent to zero sequences. Further by Lemma 5.9 and Corollary 5.10 we have that at every step: The inclusion of the additional, first step term: has clearly the effect of shifting the curve of the ( T Li (n) − π n ) significantly down, thereby upsetting the stepwise inter-relationships. However, by Theorem 5.15 we have that: with θ (n) given by the Definition 5.1. Thus, at p n = 43 the tailored integral concludes the stage of recovery instigated by the addition of the initial term, vide equation 5.55 above. The difference: at p 15 = 47 attains ∼ 0.0278753 and diverges decisively. Consequently, by Theorem 5.12 it must be true that the relation 5.57 holds for all p (n) ∈ N | p (n) ≥ 47. The difference: at p 15 = 47 the difference attains ∼ 0.00448015 and further diverges. Since the relation 5.54 holds stepwise for all p (n) ∈ N | p (n) ≥ 13, therefore, necessarily it must be true that the relation 5.58 holds for all p (n) ∈ N | p (n) ≥ 47. Consequently, for all p (n) ∈ N | p (n) ≥ 47 we obtain: Thus, concluding the proof.
Therefore on the basis of Theorems 5.16 and 5.17, the relation holds for all prime numbers p ∈ N | p ≥ 47:   The calculation of the integral T Li (n) may become computationally quite demanding for larger values of n ∈ N. This attribute of computation of the tailored integral T Li (n) pertains to the sequential, exhaustive process of calculation of the value of log p (n) ♯, which has exponential time complexity. Therefore, at the cost of an increased estimation error, this section presents a method to obtain the approximate value of the Supremum/Infimum in an efficient manner.
where the prime p n ∈ N | p n ≥ 11 Definition 5.19 (Lower Bound integration limit).
Theorem 5.20 (Upper and Lower Bounds on π n ). Let π n be the prime counting function. Then, the upper and lower bounds on π n are given by: With the constant C 1 = 0. Setting C 1 = 0.2 will include all n ∈ N | 11 ≤ n ≤ 23, it will however interfere with subsequent results.
Remark 5.8. The proof is a multistep process and will be carried over several of the following theorems/lemmas. The proof pertains to the Upper Bound only, an analogous process needs to be carried out in the case of the Lower Bound, which is left at the discretion of the reader.
By Lemma 4.5 we have: where p n is the greatest prime number p ∈ N | p ≤ n At the cost of an increased estimation error we may estimate the true value of the tailored integral quite easily.
The difference in values taken by the logarithmic integral Li (n) and the estimate of the tailored logarithmic integral U B (n) given by Theorem 5.20, for all n ∈ N | n ≥ 11, increases without a bound as n tends to infinity: → ∞ Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 Consequently, the value of the U B (n) will always remain less than the value obtained by the Gauss' Li (n) : Proof. Clearly, where p (n) is the largest prime p ≤ n. Since the limit diverges: the difference between the pertinent intervals of computation of Li (n) and U B (n) increases unboundedly. Due to the fact that, the difference between the intervals of computation increases at a rate proportional to k ( log p (n) ) 3 √ p (n) where k ≈ 0.220367 for all n ∈ N | n ≥ 94. The exact difference between the intervals of computation of Li (n) and U B (n) is given by: However, because the difference ( n − p (n) ) is bounded above by the maximal prime gaps Supremum given by Theorem 3.6, its contribution for large n is negligible. Therefore, we may drop the difference ( n − p (n) ) and by an application of the PNT we obtain an estimate of the true value of the minimum difference between the two integrals: The estimate 5.69 clearly increases monotonically without a bound. Consequently, and increases as n tends to infinity. This implies, Thus completing the proof.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2  Remark 5.9. Due to the uneven distribution of primes, both the difference n−p (n) and p (n) − log p (n) ♯ are inherently highly oscillatory. Approximating log p (n) ♯ by the difference: as the upper integration limit for the U B (n) , incorporates this effect into the estimation error of the U B (n) . Another consequence of the application of the estimates of log p (n) ♯ instead of the exact values, is that U B (n) also exhibits the tendency to follow the Gauss' Li (n) . This is clearly visible over the intervals where both π (n) and T Li (n) tend to "sag", the estimate U B (n) keeps on going relatively unaffected. Please refer to the graph 18 in the Appendix. Because U B (n) uses the value of the greatest prime p ∈ N | p ≤ n in the calculation of the upper limit of integration, as a result its graph continues in a straight level line across every prime gap. The omission of the term n − p (n) from the estimate of the difference of Li (n) and U B (n) : Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 caused the results to be accurate at the primes only; since n = p (n) at such point.
In the intermittent space, the true value of Li (n) − U B (n) increases. The greater the gap between the primes, the greater the difference between the estimate 5.71 and the true value of the difference Li (n) − U B (n) . Consequently, for a given prime gap, we have that the locally biggest difference occurs at every n ∈ N | n = p (n) − 1.
However, by Theorem 3.6, the difference n − p (n) is bound to be less than the maximal gaps Supremum: log p (n) = 5 log 10 hence the relative contribution of the difference n − p (n) decreases as n increases, consequently, gradually losing significance. In fact at 10 15 the ratio: 5 log 10 (log 10 n) thus, the error made in estimation by omission of the term (n − θ U ), is less than 0.15 percent at that point.

Theorem 5.22 (Infimum Of The Difference Li (p) − U B (p) ).
In an instance when n is a prime number, the Infimum bound on the difference Li (p) − U B (p) computed at the primes p ∈ N | p ≥ 11, is given by: In this case, the upper estimation error bound, for all p ∈ N | p ≥ 11 is given by, ] ≤ 1 (log (10) (log (p n ))) 2 + LT I (p(n)) Similarly, the lower estimation error bound, for all p ∈ N | p ≥ 263 is given by: Proof. From Definition 5.18 and Theorem 5.20, the length of the interval separating the Li (n) and U B (n) is given by: where p (n) is the biggest prime p ≤ n. When n is a prime however, ( n − p (n) ) = 0. By the application of the PNT therefore, we may approximate the true value of the Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 difference Li (p(n)) − U B (p(n)) at the primes within the range: LT I (p(n)) is a positive, monotone, increasing without a bound function. By Theorem 5.21, for all n ∈ N | n ≥ 11 the difference Li (n) − U B (n) > 0 and diverges. The Gauss' offset integral Li (n) clearly increases monotonically, as well as the bound U B (n) . Suppose that Theorem 5.22 is false, thus, in accordance with the hypothesis the difference: However, the inequality 5.75 at p n = 11 attains ∼ 0.011694 and asymptotically tends to zero as p n increases unboundedly. Therefore, it is a positive decreasing sequence of real numbers. We implement therefore the Second Ratio Test. Define the test sequence: } the Second Ratio Test, given by max The test sequence tends from below asymptotically to ∼ π 4 > 1 2 . Therefore, by the definition of the Second Ratio Test we conclude that, the series formed by the terms of 5.76 diverges as p n increases unboundedly. Consequently, we have a contradiction to the initial hypothesis. Therefore, for all p ∈ N | p ≥ 11 the relation is valid: Suppose now that ∀n ∈ N | n ≥ 347 the following inequality is false: However, the relation at p n = 347 attains ∼ −60.9818 and diverges with a rate of divergence ∝ k p n | f or k ∼ 1 for larger p n . Therefore, for all p ∈ N | p ≥ 263 define a positive valued test sequence and implement the Cauchy Root test: The Root Test, at p n = 347 attains ∼ 1.01191647 and converges asymptotically, strictly from above to 1. Consequently, by the definition of the Cauchy's Root Test, the series formed from the terms of the sequence diverges. Hence, we have a contradiction to the initial hypothesis. Consequently, ∀n ∈ N | n ≥ 347 the inequality is valid: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 Suppose now in turn, that ∀n ∈ N | n ≥ 11 the following inequality is false: ] −1 − (log (10) (log (p n ))) 2 > 0 However, inequality 5.80, at p n = 11 attains approximately 55.0282 and diverges. Therefore, define a positive valued test sequence and apply the Cauchy's Root Test: At p n = 11 the Root Test attains approximately 1.43958 and converges asymptotically, strictly from above to 1. Consequently, by the definition of the Cauchy's Root Test, the series formed from the terms of the sequence diverges. Therefore, we have a contradiction to the initial hypothesis. This implies that ∀n ∈ N | n ≥ 11 the inequality is valid: ] −1 > (log (10) (log (p n ))) 2 Hence, from the Inequalities 5.77, 5.79 and 5.81 for all p ∈ N | p ≥ 11 we have: as well as, ] ≤ 1 (log (10) (log (p n ))) 2 + LT I (p(n)) Furthermore, for all p ∈ N | p ≥ 347 we have: A straightforward computer calculation verifies that the Inequality 5.84 holds for all p ∈ N | 263 ≤ p ≤ 347 as well. Consequently, Inequality 5.84 holds for all primes p ∈ N | p ≥ 263 as stated, thus concluding the proof.
Theorem 5.22 implies that we have a very good approximation of the difference Li (n) − U B (n) at the primes within the range, hence the least difference Li (n) − U B (n) . 5.7. Supremum of π (n) and the Skewes' π (n) > Li (n) problem appraisal.
About 1792 Carl F. Gauss postulated the PNT on the basis of empirical evidence. He thought that: Gauss' belief relied on his observations made, of the tables of primes up to n = 3, 000, 000. Many of those he constructed by hand himself. His belief was Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 shared by Bernhard Riemann and indeed many other mathematicians of the 19-th century. In 1914 John E. Littlewood presented the proof that: Littlewood's proof of 1914 [25], [22], depends upon the size of the log (log (log n)) for large n. Since however the Littlewood's proof was not constructive [28], Stanley Skewes in 1933 presented a proof (assuming Riemann's hypothesis), that there exist values of n such that: π (n) > Li (n) for n ∈ N | n < 10 10 10 34 In 1955 S. Skewes re-appraised the problem, this time without the assumption of Riemann's hypothesis he produced a different bound: π (n) > Li (n) for n ∈ N | n < 10 10 10 10 3 This legendary bound has since been lowered very significantly, however, it still remains out of reach of direct verification [2]. The theory of the tailored integral developed up to this point permits us to attack Skewes' problem and to prove conclusively that Li (n) > π (n) ∀n ∈ N | n ≥ 11.

Theorem 5.23 (The Supremum Bound Of Estimation Of π (n) ).
The Tailored Integral is less or at most equal in value to the estimate U B (n) : Further, the tailored logarithmic integral T Li (n) constitutes the Supremum estimation bound of the prime counting function π (n) : consequently, Proof. By Theorem 5.21 we have that: where the difference in values taken by the estimate of the tailored logarithmic integral and the Gauss' logarithmic integral diverges as n tends to infinity: On the other hand, by Theorem 5.15 we have: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 With the estimation error increasing unboundedly as n tends to infinity: Now, by Lemmas 4.5 and 4.7, we have for all p (n) ∈ N | p (n) ≥ 2 where p n is the greatest prime number p ∈ N | p ≤ n: Which gives the interval containing the true value of θ (n) = log p (n) ♯: The upper endpoint of the interval 5.94 is the θ U . In the case that the value of log p (n) ♯ is located close to the upper endpoint of the interval 5.93, this implies that θ (n) θ U . Consequently, Hence, the Infimum of the difference U B (n) − T Li (n) is: The maximum possible value of the difference U B (n) − T Li (n) may be estimated by using the length of the interval 5.94 and applying the PNT: Clearly, the limit of the estimated maximum value of the difference 5.95 diverges, as n tends to infinity. Hence we have that: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 as well as by Theorem 5.15 we have: Since U B (n) − T Li (n) ≥ 0 for all p (n) ∈ N | p (n) ≥ 43, T Li (n) − π (n) > 0 for all n ∈ N | n ≥ 43, this implies that U B (n) − π (n) > 0 for all n ∈ N | n ≥ 43. Direct computation shows that U B (n) − π (n) > 0 for all n ∈ N | 23 ≤ n ≤ 43. Please refer to Table 4 below. Consequently, from 5.89, 5.91 and 5.96 we have that: This shows that T Li (n) constitutes the Supremum estimation bound of the prime counting function π (n) for all n ∈ N | n ≥ 43. Direct computation confirms that the difference U B (n) − T Li (n) > 0 holds for all n ∈ N | 11 ≤ n ≤ 43. Please refer to Fig. 11. Therefore, Theorem 5.23 holds as stated, concluding the proof.
Necessarily, this implies that the Littlewood's theorem of 1914 and hence the relation 5.97 above are both false, disproving them for every n ∈ N | n ≥ 11. Since the relation 5.97 is obviously false within the range n ∈ N | 2 ≤ n ≤ 11, consequently this implies that the Littlewood's theorem of 1914 is false for every n ∈ N.
Theorem 5.23 implies that Carl F. Gauss' belief, shared by Bernhard Riemann and indeed many other mathematicians of the 19-th century, was correct thereby proving their historical guess.

Estimation error bounds on the difference T Li
Because the Infimum and Supremum estimation error bounds are inherently difficult to compute for large n ∈ N, this section presents alternative lower and upper estimation error bounds. by: Where γ is the Euler-Mascheroni constant and A is the Glaisher-Kinkelin constant given by definition 1.5.

Proof.
Evidently, π (m) defines a weakly monotone, divergent function. By Theorem 5.15 the estimation error of the tailored logarithmic integral T Li (m) defines a monotone divergent sequence. Also, the lower estimation error bound LEB (m) clearly defines a monotone divergent sequence. Suppose that Theorem 5.25 is false for m ∈ N | m ≥ 1 000 000 007, then it has to be true that: Which is equivalent to say, However, at m = 1 000 000 007 the difference 5.100 attains ∼ 9.9890903 * 10 22082891 and rapidly diverges as m increases unboundedly. Therefore, the difference 5.100 generates positive numerical sequence in R. Thus, we implement the Cauchy's Root Test 1 : The test at m = 1 000 000 007 attains ∼ 2.7182926769186047 and converges to ∼ exp (1) strictly from above. By the definition of the Cauchy's Root Test this 1 The degree n of the root pertains to the prime number pn being the n-th prime number, the largest one that satisfies the relation pn ≤ m.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 implies that a series formed by the terms of the Inequality 5.100 necessarily diverges. Consequently, the difference: 2γ(log 10 m) ( for all m ∈ N| m ≥ 1 000 000 007. This implies that we have a contradiction to the hypothesis. Direct computer calculation confirms that Theorem 5.25 also holds within the range for all m ∈ N| 11 ≤ m ≤ 1 000 000 007. The pertinent data had been rendered in graphical form, please refer to Fig. 13 to 17 in the Appendix. Therefore, Theorem 5.25 holds for all m ∈ N|m ≥ 11; thus concluding the proof.

Theorem 5.26 (Upper Estimation Error Bound).
The error of estimation of the prime counting function π (m) by the application of the tailored logarithmic integral T Li (m) , for all m ∈ N | m ≥ 3 is bounded above by: Where T C is given by the Definition 1.6.

Proof.
Evidently, π (m) defines a weakly monotone, divergent function. By Theorem 5.15 the estimation error of the tailored logarithmic integral T Li (m) defines a monotone divergent sequence. Also, the upper estimation error bound UEB (m) clearly defines a monotone divergent sequence. Suppose that Theorem 5.26 is false for m ∈ N | m ≥ 1 000 000 007, then it has to be true that: Which is equivalent to say, However, at m = 1 000 000 007 the difference 5.103 attains ∼ 3.81666351 * 10 29156538 and rapidly diverges as m increases unboundedly. Therefore, the difference 5.103 generates positive numerical sequence in R. Thus, we implement the Cauchy's Root Test 2 : The test at m = 1 000 000 007 attains ∼ 2.71829300286192 and converges to ∼ exp (1) strictly from above. By the definition of the Cauchy's Root Test this implies that a series formed by the terms of the Inequality 5.103 necessarily diverges.
Consequently, the difference: ( for all m ∈ N| m ≥ 1 000 000 007. This implies that we have a contradiction to the hypothesis. Direct computation ∀m ∈ N | 3 ≤ m ≤ 1 000 000 007 verifies that Theorem 5.26 also holds within this range. Pertinent data had been rendered in graphical form, please refer to Fig. 13 to 17 in the Appendix. Therefore, Theorem 5.26 holds for all m ∈ N | m ≥ 3, concluding the proof. Remark 5.10. Both the lower and upper estimation error bound follow the T Li (n) estimation error curve very closely. This situation extends over a prolonged interval. Please refer to Table 6 in the Appendix for a listing of the local minima.

Gauss' logarithmic integral Li (n) and Riemann's hypothesis
This section develops the mathematical basis, to link unambiguously the tailored integral theory with the Gauss' logarithmic integral error term.
6.1. Divergence of the estimation error of Li (n) . Theorem 6.1 (Divergence of the estimation error of Li (n) ).
The estimation error of the Gauss' logarithmic integral Li (n) diverges as n tends to infinity:

Proof.
By Theorems: 5.15, 5.21 and 5.23 we have that: and clearly: π (n) < Li (n) ∀n ∈ N | 11 ≤ n ≤ 43 The estimation error of the tailored integral T Li (n) by Theorem 5.15 diverges: By Theorem 5.21 the difference in values between the estimate of the tailored integral and the Gauss' logarithmic integral diverges: → ∞ Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 Consequently therefore: Thus concluding the proof of Theorem 6.1.
6.2. Li (n) estimation error bounds. Figure 12. The figures show the estimation error Li n − π n (gray) and the upper and lower bounds 6.10 and 6.6 (red), the figure is drawn with respect to ξ at every n ∈ N | 11 ≤ n ≤ 3000 in left figure and n ∈ N | 11 ≤ n ≤ 300000 in right figure.
The estimation error of the tailored logarithmic integral T Li (n) produces virtually oscillation free curve which by Theorem 5.15 diverges: The smooth characteristic of the curve makes it possible to establish the estimation error bounds for the Li (n) by converting the estimation error bounds of the tailored logarithmic integral T Li (n) − π (n) to the upper and lower bounds for the estimation error of the logarithmic integral Li (n) − π (n) , by the use of a specific multiplier. The upper bound implements the multiplier: The lower bound applies the multiplier: Definition 6.3. M 2 = (log 10 m) ( √

3−1)
A(log 10) for m ∈ N Where T C and A are given by the definitions: 1.6 and 1.5 respectively, and γ is the Euler-Mascheroni constant gamma.

Theorem 6.4 (Lower Li (m) Estimation Error Bound).
For any m ∈ N | m ≥ 11, where p n is the n-th prime, the largest one satisfying the relation: p n ≤ m, the following relation holds: A (log 10) Where A is given by the definition 1.5.
The estimation error of T Li (m) as given by Theorem 5.15, and the estimation error of Li (m) as given by Theorem 6.1, both diverge as m tends to infinity. Due to the fact that the T Li (m) estimation error increases only at the primes and remains constant between them, consequently, for all m ∈ N | m ≥ 3 it defines a weakly monotone, divergent function. Clearly, π (m) defines a weakly monotone, divergent function as well. Necessarily therefore, ( (log 10 m) ( A (log 10) defines a monotone, divergent function. The Gauss' logarithmic integral Li (m) clearly is strictly monotone, divergent function. Suppose therefore, that Theorem 6.4 is false. Hence in accordance with the hypothesis we have that: A (log 10) The difference Li (m) − π m clearly is highly oscillatory, which obviously applies equally well to the difference Li (m) − T Li (m) . Therefore to smooth out the characteristics of the difference of the terms of Inequality 6.7 we take the exponential: A (log 10) ) However at p n = 11, the difference attains ∼ 153.504313 and rapidly diverges.
Since the difference produces positive numerical output, we apply the Cauchy's Root Test 3 : A (log 10) The test at p n = 47 attains ∼ 2.79234 and converges strictly from above to ∼ exp(1). By the definition of the Cauchy's Root Test this implies that a series formed by the terms of the Inequality 6.8 necessarily diverges. Consequently, the difference: A (log 10) ) for all m ∈ N| m ≥ 47. This implies that we have a contradiction to the hypothesis. Computer calculation confirms that Theorem 6.4 also holds within the range for all m ∈ N| 11 ≤ m ≤ 47. Therefore, Theorem 6.4 holds for all m ∈ N|m ≥ 11; thus concluding the proof.

Theorem 6.5 (Upper Li (n) Estimation Error Bound).
For any m ∈ N | m ≥ 3, where p n is the n-th prime, the largest one satisfying the relation: p n ≤ m, the following relation holds: Where T C is given by the definition 1.6.

Proof.
The estimation error of T Li (m) as given by Theorem 5.15, and the estimation error of Li (m) as given by Theorem 6.1, both diverge as m tends to infinity. Due to the fact that the T Li (m) estimation error increases only at the primes and remains constant between them, consequently, for all m ∈ N | m ≥ 3 it defines a weakly monotone, increasing without bound function. Clearly, π (m) defines a weakly monotone, divergent function as well, as is (log 10 m) √ 2 function. Necessarily therefore, defines a monotone, divergent function. The Gauss' logarithmic integral Li (m) clearly is monotone and divergent function. Suppose therefore, that Theorem 6.5 is false. Hence in accordance with the hypothesis we have that: The difference Li (m) − π m clearly is highly oscillatory, which obviously applies equally well to the difference Li (m) − T Li (m) . Therefore to smooth out the characteristics of the difference of the terms of Inequality 6.11 we take the exponential: However at p n = 541, the difference attains ∼ 3.57932 × 10 47 and rapidly diverges.
Since the difference produces positive numerical output, we apply the Cauchy's Root Test 4 : The test at p n = 541 attains ∼ 2.95685 and converges strictly from above to ∼ exp(1). By the definition of the Cauchy's Root Test this implies that a series formed by the terms of the Inequality 6.12 necessarily diverges. Consequently, the difference: for all m ∈ N| m ≥ 541. This implies that we have a contradiction to the hypothesis. Direct computation ∀m ∈ N | 3 ≤ m ≤ 541 verifies that Theorem 6.5 also holds within this range. Therefore Theorem 6.5 holds for all m ∈ N | m ≥ 3, concluding the proof.
Remark 6.1. Determining the Li (n) estimation error bounds by the application of T Li (n) estimation error curve is computationally very inefficient process, applicable to a relatively small values of n ∈ N only, a different more efficient method will be presented shortly.
6.3. Primary estimation error bound of Li (n) .
The equation arising in estimation of the true value of π (n) by the application of the logarithmic integral (6.14) has been shown in 1901 by Niels F. Helge von Koch, to be equivalent to the Riemann's Hypothesis. The size of the estimation error term which depends on the gaps between primes, is intimately connected with the location of the zeroes of the Riemann zeta function.
Theorem 6.6 (Primary Lower Bound Of Li (n) Estimation Error). For any n ∈ N | n ≥ 53, the error made in estimation of the true value of π (n) by the application of the logarithmic integral is bounded below by: Proof. By Theorems: 5.15, 5.21 and 5.23 we have that: The estimation error of the tailored integral T Li (n) by Theorem 5.15 diverges: By Theorem 5.21 the difference in values attained between the estimate of the tailored integral and the Gauss' logarithmic integral diverges: Both Li (n) > π (n) and T Li (n) > π (n) for n ∈ N | n ≥ 43, from 6.16 we obtain: by Theorem 5.15 we have: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 July 2021 doi:10.20944/preprints202006.0365.v2 Consequently, Direct evaluation confirms that: Please refer to the Table 7 in the Appendix. Therefore, the sum of reciprocals of prime numbers is for all n ∈ N | n ≥ 53, the primary lower bound of the estimation error, made by the application of the Gauss' logarithmic integral. Concluding the proof of Theorem 6.6.

Lower bound on the error term of the Gauss' logarithmic integral.
By Theorem 6.6 we have that the sum of reciprocals of prime numbers for all n ∈ N | n ≥ 53 is the primary lower bound on the estimation error made by the application of the Gauss' logarithmic integral. However, despite the fact that this bound may well serve its purpose, it is far from being optimal and evidently does not prove equation 6.21.

Theorem 6.7 (Lower Estimation Error Bound on The Logarithmic Integral).
For any n ∈ N | n ≥ 4, the error made in estimation of the true value of π (n) by the application of the logarithmic integral is bounded below by: A (log 10) Proof. The Theorem 6.4 states that: A (log 10) Theorem 5.25 states that the lower estimation error bound on the tailored logarithmic integral is given by: 2γ(log 10 n) ( Therefore for all n ∈ N | n ≥ 11, A (log 10) ) × ( LEB (n) ) = 1 2γ ( √ n (log 10 n) 2 + 5 log 10 (log 10 n)−6 ) − ( (exp(2)) (log 10 n) ( √
6.5. Upper bound on the error term of the Gauss' logarithmic integral.
Riemann's hypothesis is the final major objective of this research. Theorem 6.9 (The Riemann's Hypothesis).
The prime counting function π (n) is given by: Proof. By Theorem 6.8 we have that: Where: ( √ n (log 10 n) + 5 (log 10 n) (log 10) +7 ) + 3.5 Suppose that: GaussUEB (n) > √ n log n Necessarily therefore, Direct computation verifies that the ratio 6.31 attains the value ≈ 0.0551976 at n = 2 and further increases. At n = 33 it exceeds the value of 1 and decisively diverges. For all n ∈ N | n ≥ 967 the ratio 6.31 diverges at a rate exceeding log (n), while further accelerating. Thus we have a contradiction to the initial hypothesis. Since the ratio √ n log (n) GaussUEB (n) diverges, this implies that the ratio GaussU EB (n) √ n log n asymptotically converges to zero, as n increases unboundedly. Analogous numerical computation confirms that at n = 2 the difference: √ n log n − (Li (n) − π (n) ) ≈ 1.98026 and diverges. Therefore, for all n ∈ N | n ≥ 2: The drawings 13 to 17 show the graphs of the upper and lower estimation error bounds (red) and the estimation error T Li (pn) − π (pn) (blue). Due to the fact that the estimation error curve T Li (pn) − π (pn) is relatively smooth, to reduce the size of the resultant database, a technique had been implemented in construction of the figures: 13 to 16. The figures were produced by computing the distance from the curve T Li (pn) − π (pn) to each bound, at every prime within a sub-interval. Two individual points exhibiting the least distance were stored per interval. The interval widths for all n ∈ N | n ≤ 10 9 were computed in accordance with the criteria: Table 5. Interval widths specifications n Interval width < 10 3 50 < 10 5 200 < 10 6 10 3 < 10 7 10 4 < 10 8 5 * 10 4 < 10 9 10 5 < 10 10 10 6 above 10 7 The graphs had been constructed in such a way that their ranges slightly overlap. The points exhibiting locally/globally least distance from either estimation error bound are indicated on the graphs by a black dot located on the respective bound curve. Table 6, specifies the coordinates of such points.  Figure 13. The figure is drawn with respect to ξ, which gives the range n ∈ N | 11 ≤ n ≤ 7 975 013, which includes the global minimum point at n = 6 862 489, ξ ≈ 3.9532140115325856. Please refer to Table 6.