Prime numbers in logarithmic intervals

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every $\lambda>1/2$ there exists a positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda\log X]$ contains at least a prime number. As a consequence we improve Cheer and Goldston's result on the size of real numbers $\lambda>1$ with the property that there is a positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda\log X]$ contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda\log X]$ contains at least a prime number. The last application of these techniques are two theorems (the first one unconditional and the second one in which we assume the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture) on the positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda\log X]$ contains no primes.


Introduction
Let X be a large parameter, P be the set of primes and λ be a positive real number. This paper is devoted to study the distribution of primes in short intervals: in particular we will give lower bounds for the proportion of positive integers m ≤ X , or p ∈ P and p ≤ X , such that the intervals (m, m + λ logX ] or (p, p + λ log X ] contain or do not contain a prime number.
Many mathematicians studied the distribution of primes in short intervals; here we just recall some fundamental papers on this topic. Several results are formulated using the quantity In 1926 Hardy and Littlewood [13], assuming the validity of the Generalized Riemann Hypothesis, gave the first non-trivial estimate E ≤ 2/3. In 1940, Erdős [5] proved unconditionally that E < 1 and in 1966 Bombieri and Davenport [1] improved this result to E ≤ 0.46650 . . .. In 1972-73, Huxley [15,16], using a new set of weights, was able to reach E ≤ 0.44254 . . .. For all these results, a suitable modification of the argument can lead to a positive proportion result on the cardinality of the integers m ≤ X such that (m, m +λ logX ] contains at least a prime number, for any fixed λ larger than the given bound for E. In 1986 Maier used his matrix-method [18] to obtain E ≤ 0.2486 . . . but this method gave no positive proportion results.
It was just in 2005 that Goldston, Pintz and Yıldırım [10] obtained that E = 0 solving a long-standing conjecture. In fact they proved quite a stronger result: there are infinitely many i such that p i+1 − p i ≪ log p i (log log p i ) 2 .
Unfortunately, it seems that this wonderful new technique gives no positive proportion results.
In 1987 Cheer and Goldston [2,3] used the integral moments of primes over integers and a refinement of Erdős's technique to prove results of this kind. We also recall that Goldston and Yıldırım [11], in a paper published in 2007 but that was developed before [10] appeared, were able to obtain a new proof of the inequality E ≤ 1/4 with a method which gives also a positive proportion result.
Here we use a new result on integral moments of primes over primes (see Theorem 1), to prove that there exists a positive proportion of primes p ≤ X such that the interval (p, p + λ log X ] contains at least a prime number for every λ > 1/2, see Theorem 2. Even if the uniformity in λ is weaker than the one proved in Theorem 1 of Goldston and Yıldırım [11] (λ > 1/4), here we obtain an evaluation of implicit constant which will be useful in the consequences. The first of them is Theorem 3 in which we improve Cheer and Goldston's [2] result on the size of λ > 1 such that there is a positive proportion of integers m ≤ X such that the interval (m, m + λ log X ] contains no primes. The second consequence of Theorem 2 is a result concerning the moments of the gaps between consecutive primes (see Theorem 4).
Theorem 5 is about the positive proportion of primes p ≤ X such that the interval (p, p + λ log X ], where λ is "small", contains no primes and its Corollary 2 concerns the positive proportion of integers m ≤ X such that the interval (m, m + λ log X ] contains at least a prime number. Our last result (Theorem 6) slightly refines a conditional theorem of Cheer and Goldston [2] on the positive proportion of primes p ≤ X such that the interval (p, p + λ log X ], where λ > 0, contains no primes.
To be more precise in describing our results we need now to give some notation and definition. Let 1 ≤ n ≤ X /2 be an integer. We define the twin-prime counting functions as follows where p, p ′ ∈ P. Moreover we will write to denote the singular series for this problem and the twin-prime constant. Letting 1 < K ≤ X be a real number, we define Let moreover P k (y) = k ∑ r=1 k r 2 r r!y r (5) be a polynomial in y where k r denotes the Stirling number of second type defined as the number of ways to partition a set with k elements into non-empty subsets having r elements each (without counting the order of the subsets).
Recalling that π(u) is the number of primes up to u and that ψ(u) = ∑ n≤u Λ(n), where Λ(n) is the von Mangoldt function, we are now ready to state the following result about integral moments of primes over primes in short intervals. In what follows we will also denote by ε a small positive constant, not necessarily the same at each occurrence, and by ω ≥ 1 a parameter that will be useful in the applications.
where P k (y) is defined in (5).
The limitation to ω > 1 in Theorem 1 and in the following applications arises from Lemma 8 below. We are mainly interested to the case h = λ log X where λ > 0 is a constant. Letting we have the Corollary 1 Let ε > 0, ω > 1 and k ≥ 2 be an integer. Let further λ > 0 be a fixed constant. Then Denoting by |C | the cardinality of a given set C , we can now state our result on |A (K)| when K is about log X .
Theorem 2 Let ε > 0, X be a large parameter and A (K) be defined as in (3). Let further λ > 1/2 be a fixed constant. We have that where c 1 (λ) = sup ℓ∈Z; ℓ≥2 sup ω>1 ∆ ℓ,ω (λ) and Theorem 2 means that there is a positive proportion of primes p ≤ X such that the interval (p, p + λ log X ], with λ > 1/2, contains at least a prime number. Our uniformity in λ is weaker than the one in Theorem 1 of Goldston and Yıldırım [11] (λ > 1/4) but there they gave no evaluation of the implicit constant. Since in the following applications we will need this, we have to use our weaker Theorem 2.
Assuming a suitable form of the k-tuple conjecture, see, e.g., equation (18) below, it is clear that equation (25) below holds for every positive λ and with the factor λ/2 − 1/4 replaced by λ/2. Hence in this case we can replace, in the statement of Theorem 2, the condition λ > 1/2 where In fact a simpler form of the constant in Theorem 2 can be proved using ω = 2 and the Cauchy-Schwarz inequality instead of the Hölder inequality. But numerical computations proved, at least for λ ∈ (1/2, 2], that the largest constants ∆ ℓ,ω (λ) and ∆ ℓ,ω (λ) are obtained with ω ≈ 5503/5000 and ℓ = 11 in the unconditional case, and for ω ≈ 5939/5000 and ℓ = 10 in the conditional case; moreover in the following application a different optimization is needed and the form in (7) is a more flexible one and leads to better final results. To write some numerical values, we have As an application of Theorem 2 we have a result about the set B 1 (K). We improve the estimates of Theorem 3 in Cheer and Goldston [2]. We also remark that, even if in [2] Cheer and Goldston used p i ∈ (X , 2X ] while we are working with p i ∈ (0, X ], we still can compare the constants involved since the estimates have a good dependence on X . Theorem 3 Let ε > 0, X be a large parameter and B 1 (K) be defined as in (4). Then there where is a positive constant, B is defined in Lemma 2 below and ∆ ℓ,ω (ν) is defined in (7). Moreover we also have where is a positive constant and B and ∆ ℓ,ω (ν) are as before.
The best λ > 1 we are able to obtain in the previous statement is where the numerical value is obtained using the estimate of Fouvry and Grupp [6] for B = 3.454, ν ≈ 0.666856, ℓ = 12 and ω ≈ 2491/2250 in (7). Since it is not completely clear how to optimize the estimates in this theorem, it is possible that the numerical values written here and later can be further improved. We remark that Cheer and Goldston [2], in their Theorem 3, proved that λ = 1 + 1/(2B) is allowed in (10). Using the estimate for B mentioned above, this leads to λ = 1.144759 . . .. Moreover, we remark that the constant c 2 (B, λ) is larger than the one in eq. (3.3) of [2]. For example, with B = 3.454, for λ = 1 + 1/(2B) we get a gain of ≈ 6.1974568 · 10 −7 obtained for ℓ = 12, ω = 2491/2250 and ν = 0.666856 . . ..
Theorem 5 Let ε > 0, X be a sufficiently large parameter and 0 < λ < 2/B − ε where B is defined in Lemma 2 below. We have that Recalling the result in Fouvry and Grupp [6], see also the remark after Lemma 2, we can set B = 3.454 and hence Theorem 5 holds for every λ < 0.579038 . . .. Assuming that the inequality in Lemma 2 holds with the best possible value B = 1 we obtain that Theorem 5 holds for every λ < 2. As a Corollary we have Corollary 2 Let ε > 0, X be a sufficiently large parameter and λ > 0. We have that The implicit constant here is the same as in Theorem 5 for λ < 2/B − ε and it is ε otherwise.
Our last theorem is a conditional result on the cardinality of A 1 (λ log X ) when λ is not in the range described in Theorem 5.

Theorem 6 Let X be a sufficiently large parameter. For λ ≥ 2/B, where B is defined in Lemma 2 below, we assume that there exists a positive constant c
Assume further that there exists an absolute constant c 6 > 0 such that for every η > λ we have Then We remark that Heath-Brown proved that the hypothesis (15) holds under the assumption of the Riemann Hypothesis and of a suitable form of the Montgomery pair-correlation conjecture, see Corollary 1 of [14]. Theorem 6 should be compared with Theorem 5 of Cheer and Goldston [2] in which our hypothesis (14) is replaced by the stronger condition they used there. We finally remark that, using equation (31) below, we can also say that Theorem 5 of Cheer and Goldston [2] holds for λ defined in (12).

Acknowledgements.
We would like to thank Professors Heath-Brown and Maier for their insights and Professor Perelli for an interesting discussion about his joint papers with Salerno.

Main Lemmas
We will use two famous results by Bombieri and Davenport.

Lemma 2 (Theorem 2 of Bombieri-Davenport [1])
There exists a positive constant B such that, for any positive ε and for every positive integer n, we have where Z(X ; 2n) and S(n) are defined in (1)-(2), provided X is sufficiently large.
Chen [4] proved that B = 3.9171 can be used in Lemma 2. Wu [22] recently slightly improved this by proving that B = 3.91045 is admissible for every value of n. For n ≤ log A X , where A > 0 is an arbitrary constant, the best result is B = 3.454 by Fouvry and Grupp [6]. Moreover we remark that similar results hold for Z 1 (X ; 2n), defined in (1), since Z 1 (X ; 2n) ≤ Z(X ; 2n) ≤ Z 1 (X ; 2n) log 2 X and hence, using the inequalities we also obtain Concerning the summation of the singular series of the twin-prime problem, we will use the following result. [7], eq. (1.13)) Let X ≥ 2. We have

Proof of Theorem 1 and Corollary 1
We now define two different averages for primes in short intervals we will need to prove Theorem 1. Let be the Selberg integral and its discrete version. In the proof of Theorem 1 we will follow the line of Perelli and Salerno [19,20] to connect the moments over primes with the corresponding ones over integers. To this end we now need several lemmas. We assume implicitly that X is sufficiently large.
Proof. The proof follows immediately inserting the following sieve estimate of Klimov [17] in Gallagher's argument (see also Theorem 5.7 of [12]) where h 1 , . . ., h r are distinct integers such that 0 ≤ h i ≤ h for every i = 1, . . . , r, and ν p (h 1 , . . ., h r ) is the number of distinct residue classes modulo p the h i , i = 1 . . . , r, occupy.
The case k = 4 of Lemma 4 was recently proved by Goldston and Yıldırım, see eq. (7.32) of [11]. We also remark that, assuming the k-tuple conjecture in the form where h 1 , . . . , h r are distinct integers and S(h 1 , . . ., h r ) is defined in (17), Gallagher [8] proved that Lemma 4 holds with the term P k (h/ log X ) replaced by P k (h/ logX ) where Now we see two lemmas on the connections between the Selberg integral and its discrete version.

Lemma 5
Let h be an integer, 1 ≤ h ≤ X . Let further k ≥ 2 be an integer. Then h). By the Brun-Titchmarsh inequality we have J k (X , h) = J k (⌊X ⌋, h) + O h k log k X (log(2h)) −k and the Lemma follows.

Lemma 6
Let ε > 0 and h ∈ R \ Z with 1 < h ≤ X . Let further k ≥ 2 be an integer. Then where P k (y) is defined in (5).
Let now u be a positive real number and This function can be easily connected with the Selberg integral. [19,20]) Let 1 ≤ h ≤ X , and k ≥ 2 be an integer. Then

Lemma 7 (Perelli-Salerno
Proof. Let N = max(m i ) and n = min(m i ) in (20). Expanding the k-th power in J k (X , h), we have say. By the partial summation formula we immediately get Σ 1 = R h 0 ψ k (x, u)du and, by the Brun-Titchmarsh inequality, we have Σ 2 ≪ h k+1 log k X (log(2h)) −k . Lemma 7 follows. The next lemma gives an upper bound for ψ k (X , h) in terms of the discrete Selberg integral.

Lemma 8
Let ε > 0, ω > 1 and 1 ≤ h ≤ X . Let further k ≥ 2 be an integer. Then where O ′ means that this error term is present only if h ∈ Z and P k (y) is defined in (5).
Proof. Since ψ k (X , u) is a positive and increasing function of u, it is easy to see that where ω > 1 is a constant. By Lemmas 5, 6 and 7, we have and hence Lemma 8 follows.
The following last lemma is a lower bound for ψ k+1 (X , h) in terms of a weighted form of the discrete Selberg integral.
Proof. First of all we remark that Recalling now that N = max(m i ) and n = min(m i ) in (20), we trivially have and Lemma 9 follows immediately combining (21) and (22). Now we are ready to prove Theorem 1 and Corollary 1. By Lemmas 9, 8 and 4 the upper bound in the statement of Theorem 1 holds for the function (log X ) −1 ∑ m≤X Λ(m)(ψ(m + h) − ψ(m)) k . It is easy to see that the contribution of m = p α with α > 1 in the previous sum is negligible. Theorem 1 hence follows by the partial summation formula since f (X ) ≤ h ≤ X 1−ε , where f (X ) → +∞ arbitrarily slowly as X → +∞. The first part of Corollary 1 can be obtained inserting h = λ log X in Theorem 1 while the second part follows from the first one using the partial summation formula.