The distribution of primes in a short interval

This research paper begins the presentation, with the topic of the distribution of primes in a short interval. The lower and upper limits for the number of primes within the interval are defined unambiguously. This provides us with a solid foundation, to resolve conclusively the Second Hardy-Littlewood´s conjecture. The paper concludes with the Merit of a Prime Gap and the Second Harald Cramer´s conjecture.


Preliminaries
Within the scope of the paper, prime gap of the size g ∈ N | g ≥ 2 is defined as an interval between two primes p n , p (n+1) , containing (g − 1) composite integers. Maximal prime gap of the size g, is a gap strictly exceeding in size any preceding gap. All calculations and graphing were carried out with the aid of the M athematica software. For all n ∈ N | n ≥ 8, we make the following definitions:

Preliminaries.
The main result of this section is the Maximum bound for the number of primes within the interval G (n) . First we define several related concepts, in accordance with the definition 1.3 of Gs (n) and 1.4 of the interval end-point t. Now, by Theorem 3.1 we have that: (2.1) π (t) > π (n) ∀n ∈ N | n ≥ 11 Theorem 3.18 states that, 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 Figure 1. shows the graphs of the actual counts of the number of primes within the interval Gs (n) (Grey), the Maximum and the average bounds (red), as well as the Infimum bound (Blue). The graphs are drawn with respect to ξ at every n ∈ N within the range, which corresponds to 8 ≤ n ≤ 3 000 in the left figure, and 8 ≤ n ≤ 10 6 in the right figure. for all p i ∈ N | p i ≥ 13: where p (i+1) is associated with the upper limit of integration θ 2 . Consequently, over the length of the interval Gs (n) we obtain: Theorem 3.21 states that ∀n ∈ N | n ≥ 13, the sum of all error terms made at every step of estimation of π (n) diverges. In particular, it states that: On the basis of Theorem 3.18 define ∀n ∈ N | n ≥ 983 the estimation error: .
and the residual estimation error: Definition 2.2 (Residual error over Gs (n) ).
By Theorem 3.21 therefore: Thus, the estimation error over the interval Gs (n) equals: Due to the finite and very short length of the interval Gs (n) the limit is: .5 implies that the largest estimation error EE (t) occurs for low n ∈ N.

The Maximum bound for the number of primes in Gs (n) .
This section establishes the Maximum bound for primes within the interval Gs (n) . The Infimum for primes in the interval Gs (n) is given by Theorem 3.2. The average number of primes within the interval is given by the PNT: Gs (n) log (t) = 5 (log 10 n) 2 − 15 8 (log 10 n) log ( n + 5 (log 10 n) 2 − 15 8 (log 10 n) ) ∀n ∈ N | n ≥ 11 Montgomery and Vaughan [26] in 1973, using the Large Sieve proved that: Paul Erdös [10] in his 1980 paper, raised the problem of the number of primes within a short interval, and he conjectured that the number of primes within a short interval may be estimated by using: Conjecture 2.3 (Primes In Short Intervals -Paul Erdös).
Where b is sufficiently large absolute constant.
Substituting into the inequality 2.8; y = n, x = n + c, b = 2 we obtain: )) Inequality 2.9, clearly implements the symmetry argument based on the PNT (equation 2.6). We assert, that the Maximum for primes within the interval c = Gs (n) is given by Theorem 2.4.

Theorem 2.4 (Maximum For Primes
In Gs (n) ). The maximum number of primes within the interval Gs (n) from n to t is given by: (2.10) Proof.
Since, (2.11) )) 10 log 10 In fact, the difference of terms of 2.11 i.e. LHS less RHS, at n = 11 attains approximately −0.296956805. The limit of the difference: )) − 10 log 10 tends asymptotically towards zero. Thus obviously, MAX (t) diverges as n increases unboundedly. The tailored integral T Li (n) and the sum ∑ Suppose that Theorem 2.4 for n ∈ N | n ≥ 983 is false. This implies that, Clearly, Hence, substituting the RHS of 2.13 into 2.12, we obtain in accordance with the hypothesis: (2.14) 2 × ( 5 (log 10 n) 2 − 15 8 (log 10 n) ) log ( n + ( 5 (log 10 n) 2 − 15 8 (log 10 n) However, at n = 983 the inequality 2.14 attains ∼ 18.696357205817502 and increases as n increases unboundedly. Since inequality 2.14 generates a positive sequence of values, we implement the Cauchy Root Test: The Root Test, at n = 22 attains ∼ 1.1736950736631349, at n = 983 the test already descended to ∼ 1.017797047552908 and continues to converge 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 2.14 diverges. Therefore, we have a contradiction to the initial hypothesis. Hence, ∀n ∈ N | n ≥ 983 the inequality is valid: Consequently, for all n ∈ N | n ≥ 983 Theorem 2.4 holds. For all n ∈ N | 11 ≤ n ≤ 983, direct evaluation evidently confirms that inequality 2.14 holds in this range as well. Please refer to Figure 2. Hence, Theorem 2.4 holds ∀n ∈ N | n ≥ 11. Figure 2. shows the graphs of inequality 2.14 within the interval at every n ∈ N| 11 ≤ n ≤ 91 in the left figure, and 11 ≤ n ≤ 8 000 in the right figure.

Corollary 2.5 (Summary Of The Distribution Of Primes In Gs (n) ).
On the basis of Theorem 3.2 and Theorem 2.4 the density of prime numbers within the interval Gs (n) , from n to ( n + Gs (n) ) for all n ∈ N | n ≥ 11 has the following bounds: with the average:

Historical reflection. Density of primes in a short interval.
Atle Selberg [35] researched the problem of the density of primes in small interval: where Φ n is positive, increasing function and for n > 0: Assuming Riemann's hypothesis Selberg was able to prove that: Over the years questions were posed as to the nature of the function Φ n . In 1985 H. Maier [25] proved that: as well as: The theory presented thus far, makes it possible to solve these problems on the interval Gs (n) . Theorem 3.2 gives the maximum gaps standard. Clearly, Gs (n) is a positive, increasing function s.t.
From the Corollary 2.5 which specifies the distribution of primes in the interval Gs (n) for all n ∈ N | n ≥ 11 we substitute therefore into 2.18 obtaining: Similarly substituting into 2.19 we obtain: Clearly corroborating Maier's results. For graphical presentation of Equations 2.20 and 2.21 please refer to Fig. 3. The relation 2.17 then becomes: Which leads to the next section's problem: the Second Hardy and Littlewood's conjecture.
figures are drawn at every n ∈ N | n ≥ 8 within the range.

The Second Hardy and Littlewood's conjecture.
In 1923 Godfrey H. Hardy and John E. Littlewood postulated a conjecture on the basis of a limited empirical evidence: Although some believe that the conjecture may be incorrect, yet no one has actually proven it to be true or false. In this section a proof is presented which establishes the truth of the conjecture 2.22 on the interval Gs (n) . Theorems: 3.2 and 2.4 make it possible to verify and prove the conjecture.

Theorem 2.6 (Hardy and Littlewood Conjecture).
The conjecture postulated by G. Hardy and J. Littlewood is valid on the interval Gs (n) given by Theorem 3.2: Proof.
By PNT the number of primes within the interval fixed at 1 and terminating at Gs (n) is asymptotic to: Rosser and Schoenfeld [32] in 1962, proved for all n | n > 10 that: The Maximum for the number of primes within the interval Gs (n) from n to t is given by Theorem 2.4: )) The ratio of the two estimates is given by: Thus, for all n ∈ N | n ≥ 401, the limit of the ratio 2.24 is given by: At n = 401 the ratio 2.24 attains the value ≈ 3.33333 and increases as n increases unboundedly. This implies that Theorem 2.6 holds for all n ∈ N | n ≥ 401. For all n ∈ N | 7 < n ≤ 401 direct computation verifies that the actual counts π (n+Gs(n)) − π n ≤ π (Gs(n)) , consequently Theorem 2.6 holds in this range. Please refer to Fig.   4. Therefore Theorem 2.6 holds ∀n ∈ N | n ≥ 7. This concludes the proof of Theorem 2.6. Inequality 2.23 compares the number of primes contained within two intervals of equal length which are some distance apart. The function π Gs (n) counts the number of primes within the interval beginning at 1 and terminating at Gs (n) . At the same time π (n+Gs(n)) − π n counts the number of primes within the interval of precisely the same length, yet beginning at n and terminating at t = n + Gs (n) . Figure 4. The drawings show the graphs of π Gs (n) (Red), the Maximum for primes within the interval Gs (n) (Blue) and the difference π (n+Gs(n)) − π n (Grey). The figure shows the last intersection of the Maximum with π Gs (n) at ξ ≈ 0.886048, which corresponds to x ∈ R | x ≈ 400. π Gs (n) exceeds the Maximum from n = 401 onwards. The L.H. figure is drawn w.r.t. ξ, at all n ∈ N | 7 ≤ n ≤ 1 500; while the R.H. figure is drawn at all n ∈ N | 11 ≤ n ≤ 750 000. The merit of a prime gap is defined as the ratio of the size of a gap beginning at p n and log (p n ). It indicates the relative size of a given gap as compared to the average prime gap about p n as given by the PNT.

Theorem 2.7 (Prime Gap Merit).
The maximal prime gap merit diverges as p n ∈ N increases unboundedly: Theorem 3.2 gives us the Supremum for the maximal prime gaps, this means that for all p n ∈ N | p n ≥ 11, we have: Consequently, the Supremum on the prime gap merit is given by: Further, for all n ∈ N | n ≥ 3 we have that: Consequently we have that: (log p n − 1) ∑ p≤n log p = (log p n − 1) (log p n ♯) < (log p n − 1) (p n ) By Theorem 3.9 and Lemmas 3.10 through 3.12 we have that for all p n ∈ N | p n ≥ 11: where p n is the biggest prime p ≤ n. H. Cramér in his paper [4] showed that for any ϵ > 0 the relation: ) conditionally holds providing that the Riemann's Hypothesis is true. This section provides an unconditional proof of the relation 2.36.

Theorem 2.8 (Sum Of Squares Of M pn Terms).
For m, n ∈ N | n ≥ 13 and p n the largest prime p ≤ n, the relation holds: Proof. For p m ∈ N | p m ≥ 13 by Theorem 3.1 and Lemma 2.7 we have: Therefore, from inequalities 2.27, 2.35 and 2.38 for all n ∈ N | n ≥ 13 At p = 13:   [8] , Some unsolved problems, Publications Of The Mathematical Institute Of The Hungarian Academy Of Sciences (1961). [9] , A survey of problems in combinatorial number theory, Annals Of Discrete Mathematics (1980

Appendix
This section contains the theory cited in the text, proofs of which are given in the provided references. Theorems 3.1 and 3.2 are stated in a slightly relaxed form, after the Floor function had been dropped: Theorem 3.1 (Maximal Prime Gaps Bound (Feliksiak [11])).
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:

Theorem 3.2 (Maximal Prime Gaps Supremum Bound (Feliksiak [12])).
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: 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: (3.6) log p (n) ♯ ∼ p (n) Lemma 3.7 (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.
The Supremum of the sum of (log p) 2 terms, for all p n ∈ N | p (n) ≥ 3 is given by: is the Golden Mean constant. The relative error of the Infimum for all p n ∈ N | p (n) ≥ 3 has the following bounds: , β2 = (log 2 + 1) and GM = ( √

Lemma 3.11 (Upper bound on the sum
The Upper bound on the sum ∑ p≤n (log p) 2 for all p n ∈ N | p (n) ≥ 11 is given by: (3.11) U B pn = (log p n − 1) p n + 1 Lemma 3.12 (Lower bound on the sum ∑ p≤n (log p) 2 ).
Lemma 3.17 (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: (3.14) for all p ∈ N | p ≥ 13 with θ 1 and θ 2 given by the Definitions 3.13 and 3.14 respectively.

Theorem 3.18 (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.
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. Definition 3.19 (Theta applicable for summation). θ 1 = log ( p (2+(k−1)) ♯ ) Definition 3.20 (Theta applicable for summation). θ 2 = log ( p (2+k) ♯ ) Theorem 3.21 (T Li (n) Estimation Error Divergence). 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 3.19 and 3.20 respectively. Besides, the prime number p (n) is defined as being the biggest prime p ≤ n. Furthermore, Regarding the proofs of Lemma 3.17 and Theorems 3.18 and 3.21, please consult Feliksiak [11].