Maximal prime gaps bounds

This paper presents research results, pertinent to the maximal prime gaps bounds. Four distinct bounds are presented: Upper bound, Infimum, Supremum and finally the Lower bound. Although the Upper and Lower bounds incur a relatively high estimation error cost, the functions representing them are quite simple. This ensures, that the computation of those bounds will be straightforward and efficient.
 The Lower bound is essential, to address the issue of the value of the lower bound implicit constant C, in the work of Ford et al (Ford, 2016). The concluding Corollary in this paper shows, that the value of the constant C does diverge, although very slowly. The constant C, will eventually take any arbitrary value, providing that a large enough N (for p <= N) is considered.
 The Infimum/Supremum bounds on the other hand are computationally very demanding. Their evaluation entails computations at an extreme level of precision. In return however, we obtain bounds, which provide an extremely close approximation of the maximal prime gaps. The Infimum/Supremum estimation error gradually increases over the range of p and attains at p = 18361375334787046697 approximately the value of 0.03.


Upper Bound on Maximal Prime Gaps from Historical Perspective
The topic of the upper bound on the maximal prime gaps, had been discussed in Feliksiak [14] research paper, where the Upper Bound on the Maximal Prime Gaps is shown to be: The Upper Bound on Maximal Prime Gaps, for all p n ∈ N | p n ≥ 11 is given by: Proof.
However, already in 1936 a Swedish mathematician Harald Cramér, attempted to formulate the maximal prime gaps upper bound, by implementing a sophisticated heuristic argument. He argued, that there exists at least one prime p ∈ N, within the interval: From the Cramér's model we have: H. Cramér asserted that α = 1. Over the years, the Cramér's Maximal Gaps Conjecture had been considered too strong. In 1995 A. Granville proposed, that there is a bound M for which: With the limit expected to be: The Infimum/Supremum bounds on the maximal prime gaps, implement Riemann Sums with regular subdivisions of 250 000 per unit. This means that over the prime gap, there are g × 250 000 subdivisions. This was done to overcome technical issues, while evaluating the definite logarithmic integral. All necessary computations were carried out with the aid of M athematica software, with numerical precision set to 300 000 decimal places.  The Upper/Lower bounds shown in Figure 1, are given by Theorem 1.1 and Theorem 3.4 respectively. The Auxiliary Infimum/Supremum bounds, presented in the Figure 1, were generated purely for visualization purposes. They are the direct consequence of Theorems 2.6 and 2.7, which specify bounds that are profoundly superior to them. Since for technical reasons, it is impossible to draw the Infimum/Supremum bounds given by Theorems 2.6 and 2.7 vs the Maximal Prime Gaps curve, consequently, only the Infimum/Supremum estimation error graphs are presented. Please refer to Figures 2 and 4. The Auxiliary Infimum/Supremum graphs presented in the Figure 1, can be generated by implementing the Riemann Sum 2.1: With the constants (Auxiliary Infimum) defined by: 1 000 000 , α = 1, β = 1.020 and C 2 = (−1) With the constants (Auxiliary Supremum) defined by: First, we make a few general definitions valid within the entire section. In this section, the Riemann sums are implemented, with regular partitioning of the maximal prime gap interval. The number of regular divisions, between any two consecutive integers equals 250 000. Maximal Prime Gap of a size g, is a gap that strictly exceeds in size, any prime gap that precedes it. Let p i and p c denote the initial prime at which the Maximal Prime Gap begins and the concluding prime at which the Maximal Prime Gap ends, respectively.  The Infimum Bound on the Maximal Prime Gaps, for p i ∈ N | p i ≥ 23 is given by: With the constants: 1 000 000 , α = 1.00187651 and β = 1.00187877 Proof.
The Riemann sum 2.2 approximates the logarithmic integral over the maximal prime gaps: Let's denote a maximal prime gap: In 1931 Westzynthius [31] proved, Equivalently, in the case of the maximal prime gaps: By the PNT for all p i ∈ N | p i ≥ 23 we may estimate the logarithmic integral, hence the Riemann's Sum: Consequently, we have: Suppose that Theorem 2.6 is false for some p i ∈ N | p i ≥ 4 652 353. Define now: The natural logarithm function (log log log p i ) /330 for p i ≥ 4 652 353, is a positive, increasing function, which at p i = 4 652 353 attains approx. 0.00304478 and diverges. Thus, in accordance with the hypothesis, at the maximal gaps we must have: Since MPG pi < g ∀p i ≥ 4 652 353, necessarily therefore: Thus necessarily, the ratio a j must be: where j is the Maximal Gap ordinal number. The ratio at j = 20, p i = 4 652 353, attains approx. 1.01267 and strictly increases at every step j, diverging as j increases. Please refer to Figure 3. Consequently this implies, that we have a contradiction to the initial assumption. This implies that:

Theorem 2.7 (Maximal Prime Gaps Supremum).
The Supremum Bound on the Maximal Prime Gaps, for p n ∈ N | p n ≥ 155 921 is given by: Figure 4. This log-linear drawing shows the graph of the estimation error made by the Supremum Bound. Please refer to Table 3 in the Appendix. The figure is drawn w.r.t. p n ∈ N, in the range 155 921 ≤ p n ≤ 18 361 375 334 787 046 697.
With the constants: 1 000 000 , α = 1.00187651 and β = 1.00187874 Remark 2.2. The range of the Supremum Bound may be extended, to include all primes p n ≥ 523 by adding the constant 0.0592 and re-computing pertinent data.

Proof.
The Riemann sum 2.14 approximates the logarithmic integral over the maximal prime gaps: Let's denote a maximal prime gap: Equivalently, in the case of the maximal prime gaps: By the PNT for all p i ∈ N | p i ≥ 155 921 we may estimate the logarithmic integral, hence the Riemann's Sum: Consequently, we have: Suppose that Theorem 2.7 is false for some p i ∈ N | p i ≥ 1 357 201. Define now: (2.20) The natural logarithm function (log log log p i ) /380 for p i ≥ 1 357 201, is a positive, increasing function, which at p i = 1 357 201 attains approx. 0.0025623 and diverges. Thus, in accordance with the hypothesis, at the maximal gaps we must have: Since MPG pi > g ∀p i ∈ N | p i ≥ 1 357 201, necessarily therefore: Thus necessarily, the ratio must be: where j is the Maximal Gap ordinal number. The ratio at j = 18, p i = 1 357 201 attains approx. 1.01717452 and strictly increases at every step j, diverging as j increases. Please refer to Figure 5. Consequently this implies, that we have a contradiction to the initial assumption. This implies that:

Lower Bound on Maximal Prime Gaps with Historical Background
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: This was the first notable result in this area. In 1931 Westzynthius [31] proved the normalized prime gap limit superior: The result 3.1 had been improved upon in 1935 by Paul Erdös [10] who proved that for c > 0 we have: Robert Rankin [24] in 1938 proved that for c > 0 we have: In a more recent development, Kevin Ford Et Al. [16] in 2016 established for all p (n+1) ∈ N | p (n+1) ≤ N , a bound of the form: with the constant c being implicit. For clarity of presentation, some graphs implement a variant of a logarithmic scaling of the horizontal axis, given by: The maximal prime gaps Lower Bound, for all p (n) ∈ N | p (n) ≥ 23 is given by: Where α and β are given by the Definitions 3.2 and 3.3 respectively, γ is the Euler-Mascheroni constant and M is given by: Proof. Theorem 1.1 gives for all p n ∈ N | p n ≥ 11, the upper bound on the maximal prime gaps: Clearly, for all p n ∈ N | p n ≥ 23: Suppose that Theorem 3.4 is false, for p n ∈ N | p n ≥ 155 921. Consequently, This implies that: Cauchy Root Test at j = 13, p n = 155 921: Where j is the Ordinal Number of the maximal gap and p n is the prime at which the maximal gap commences. Furthermore, the Cauchy's Root Test at every step j exceeds 1, while decreasing strictly from above towards 1. Please refer to Figure 6. Hence, by the definition of the Root Test, the sequence |a j | diverges. In accordance with the initial hypothesis, this implies that: However, at p n = 155 921 the difference 3.13 attains approx. 698.282 and rapidly diverges. Please refer to Figure 6. Hence, we have a contradiction to the initial assumption, necessarily this implies that for any p n ≥ 155 921:   The lower bound on Maximal Prime Gaps, presented by Ford et al [16], attains values that are less than zero over the range of the maximal prime gaps for p (n+1) ∈ N | 11 ≤ p (n+1) ≤ 2 010 881. Please refer to Table 5 in the Appendix. Consequently, the log-log figures can only be drawn for p (n+1) ∈ N | p (n+1) ≥ 4 652 507. Pragmatically, we prove Theorem 3.5, beginning at the lowest C value. This means, at the prime gap terminating at p (n+1) = 19 581 334 193 189. The value of the constant C is only an approximation, as it is computed w.r.t. the Lower Bound, given by Theorem 3.4. The approximation is however sufficient and suitable, to compute the approximate value of C, at any technically feasible (or desired) point n ∈ N | n ≥ 4 652 507. The primary lower bound for all p (n+1) ∈ N | p (n+1) ≥ 4 652 507 (Ford et al [16]), is given by: The implicit constant C can be estimated by implementing C LB p (n+1) /PB p (n+1) . The value of C, for p (n+1) ≥ 4 652 507, is C ≥ 32, and gradually increases unboundedly as p (n+1) tends to infinity.
Remark 3.1. Corollary 3.5 is a direct consequence of Theorem 3.4. Hence, a simple validation is sufficient.

Proof.
By Theorems 1.1 and 3.4 we have: Kevin Ford Et Al. [16] have shown, that for any sufficiently large p (n+1) ≤ N : The difference: At p (n+1) = 19 581 334 193 189 attains 552.277785 and increases. Please refer to Figure 7 and Table 5. Implementing Cauchy Root Test: where j ∈ N | j ≥ 55 is the ordinal number of the maximal prime gap. At p (n+1) = 19 581 334 193 189 the Cauchy Root Test attains approx. 1.06525 and converges asymptotically, strictly from above to 1. Please refer to Table 5. Consequently, by the definition of the Cauchy Root Test, the ratio C j diverges for p (n+1) ≥ 19 581 334 193 189. This implies that the difference 3.18 also diverges. The ratio  The graphs in the Figure 9 have been generated to simply assist in visualization, of the C constant's slow rate of divergence and/or convergence of the Cauchy Root Test. Although the figures are not a part of the verification itself, they aid to understand the concept and help to shape up our expectations.