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Primes in tuples. I. (English) Zbl 1207.11096
These papers (Part II has been published in Acta Math. 204, No. 1, 1–47 (2010)) present a breakthrough in our understanding of differences between consecutive primes. If \(d_n=p_{n+1}-p_n\) is the difference between consecutive primes then it follows from the Prime Number Theorem that \(d_n/\log p_n\) has average 1. It has been known since the work of Westzynthius in 1931 that \(d_n/\log p_n\) can be arbitrarily large. In contrast, until the present papers the problem of small values had been a well-known open question, being a weaker version of the problem about bounded gaps \(d_n\) or even twin primes, for which \(p_{n+1}-p_n=2\).
The main results of the first paper are on the one hand that \[ \liminf_{n\rightarrow\infty}\frac{p_{n+1}-p_n}{\log p_n}=0 \tag{\(*\)} \] and on the other that, if a certain improved version “\(\text{BV}(\theta)\)” of the Bombieri–Vinogradov Theorem holds, then \[ \liminf_{n\rightarrow\infty}p_{n+1}-p_n<\infty.\tag{1} \] The hypothesis \(\text{BV}(\theta)\) is that \[ \sum_{q\leq x^{\theta}}\max_{(a,q)=1} \left|\psi(x;q,a)-\frac{x}{\phi(q)}\right|\ll_A x(\log x)^{-A}\tag{2} \] for any constant \(A\), and what is required is that this should hold for some \(\theta>1/2\). The Bombieri–Vinogradov Theorem itself allows any \(\theta<1/2\), so that the smallest improvement would suffice to deduce (1). The Elliott–Halberstam Conjecture would similarly permit us to take any \(\theta<1\), which would be more than sufficient for (1). The paper makes a precise connection between the admissible value of \(\theta\) and the size of \(\liminf_{n\rightarrow\infty}p_{n+1}-p_n\), and shows for example that under the Elliott–Halberstam Conjecture one has \[ \liminf_{n\rightarrow\infty}p_{n+1}-p_n\leq 16.\tag{3} \] The second paper proves a stronger version of (\(*\)), namely that \[ \liminf_{n\rightarrow\infty}\frac{p_{n+1}-p_n}{(\log p_n)^{1/2}(\log\log p_n)^2}<\infty.\tag{4} \] The methods of these papers are also partially successful with \(p_{n+v}-p_n\) for values of \(v\geq 2\). It is shown that under the hypothesis \(\text{BV}(\theta)\) one has \[ \liminf_{n\rightarrow\infty}\frac{p_{n+v}-p_n}{\log p_n}\leq (\sqrt{v}-\sqrt{2\theta})^2\tag{5} \] so that the Elliott–Halberstam conjecture would imply that \[ \liminf_{n\rightarrow\infty}\frac{p_{n+2}-p_n}{\log p_n}=0. \] Unconditionally the \(\liminf\) in (5) is \(\leq(\sqrt{v}-1)^2\), but it is stated that the techniques of the paper may be combined with H. Maier’s matrix method [Mich. Math. J. 35, No. 3, 323–344 (1988; Zbl 0671.10037)] to show that \[ \liminf_{n\rightarrow\infty}\frac{p_{n+v}-p_n}{\log p_n}\leq e^{-\gamma}(\sqrt{v}-1)^2, \] where \(\gamma\) is Euler’s constant. In particular one would have \[ \liminf_{n\rightarrow\infty}\frac{p_{n+2}-p_n}{\log p_n}\leq 0.096\ldots \] The most obvious outstanding question is whether one might use an improved version of the Bombieri–Vinogradov Theorem (along the lines of the results in [E. Bombieri, J. B. Friedlander and H. Iwaniec, Acta Math. 156, 203–251 (1986; Zbl 0588.10042)] for example) to establish (1). Unfortunately it seems that the method of the present papers needs a maximum over many values of \(a\) in (2), while known sharpenings of the Bombieri–Vinogradov Theorem require inconvenient restrictions on \(a\).
The basic approach in the papers is motivated by Selberg’s attack [Collected Papers. Vol. II, Springer-Verlag, New York (1991; Zbl 0729.11001)] on almost-primes \(n(n+2)\), and by the reviewer’s extension [Mathematika 44, No. 2, 245–266 (1997; Zbl 0886.11052)] to general \(k\)-tuples. In the latter, one examines \(k\)-tuples \((n+h_1,\ldots,n+h_k)\) and compares the sums \[ S_1=\sum_{N<n\leq 2N}\left\{\sum_{j\leq k}d(n+h_j)\right\} \left(\sum_{d|\prod_{j\leq k}(n+h_j)}\lambda_d\right)^2 \] and \[ S_2=\sum_{N<n\leq 2N}\left(\sum_{d|\prod_{j\leq k}(n+h_j)}\lambda_d\right)^2 \] for suitable sieve weights \(\lambda_d\). If one can show that \(S_1<HS_2\) for some integer \(H\) then there must be an \(n\) with \[ \sum_{j\leq k}d(n+h_j)\leq H-1. \] In the first paper the sum \(S_1\) is replaced by \[ S_1=\sum_{N<n\leq 2N}\left\{\sum_{j\leq k}\theta(n+h_j)\right\} \left(\sum_{d|\prod_{j\leq k}(n+h_j)}\lambda_d\right)^2, \] where \(\theta(m)=\log m\) for \(m\) prime, and \(=0\) otherwise. Then if \(S_1>(\log 2N)S_2\) there must be an integer \(n\in (N,2N]\) with \[ \sum_{j\leq k}\theta(n+h_j)> \log 2N, \] whence there are at least two primes in the \(k\)-tuple \((n+h_1,\ldots,n+h_k)\). Naturally, the method is doomed to failure unless for every prime \(p\) there is an \(n\) such that none of \(n+h_1,\ldots,n+h_k\) are divisible by \(p\). One says that a \(k\)-tuple which is not excluded in this way is “admissible”.
One therefore seeks to choose coefficients \(\lambda_d\) for \(d\leq R\) say, so as to maximize \(S_1/S_2\). At present we do not know how to do this. However to minimize \(S_2\) one could choose weights close to \(\lambda_d=\mu(d)(\log^+ R/d)^k\), and the authors work with the related choice \[ \lambda_d=\lambda_d^{(k,l)}=\mu(d)(\log^+ R/d)^{k+l}, \] where \(l\) is an integer parameter to be chosen later. Indeed one might use \(\mu(d)(\log^+ R/d)^kW(R/d)\) for a general weight function \(W\), but it appears that \(W(x)=x^l\), for a suitable \(l\), is not far from optimal.
One then has the task of evaluating the sums \(S_1\) and \(S_2\). Taking \(k\) and \(l\) to be fixed, and \(0\leq h_i\leq N^{1/5}\) say, one can achieve this providing that \(\text{BV}(\theta)\) holds, and \(R= N^{\theta/2-\delta}\) for some fixed \(\delta>0\). The techniques used here are related to those in earlier work by the first and third authors, see [Integers 3, Paper A05, 66 p., electronic only (2003; Zbl 1118.11039)] and [Proc. Lond. Math. Soc. (3) 95, No. 3, 653–686 (2007; Zbl 1134.11034)].
The outcome is that, for an admissible \(k\)-tuple one has \[ \frac{S_1}{S_2\log 2N}\rightarrow \frac{2\theta k}{k+2l+1}\frac{2l+1}{2l+2}, \tag{6} \] so that a suitable choice for \(k\) and \(l\) will yield (1) providing that \(\theta>1/2\). However when \(\theta=1/2\) one produces bounds approaching 1 from below. To achieve (3) one takes \(k=6\) and works with the admissible 6-tuple \((n,n+4,n+6,n+10,n+12,n+16)\). This however requires a slight modification in the choice of sieve weights, and the authors use a linear combination \(\lambda_d^{(6,0)}+c\lambda_d^{(6,1)}\).
To prove (4) one may modify \(S_1\), replacing it by the larger sum \[ S_1'=\sum_{N<n\leq 2N}\left\{\sum_{h=1}^L\theta(n+h)\right\} \left(\sum_{d|\prod_{j\leq k}(n+h_j)}\lambda_d\right)^2, \] where \(L\) tends to infinity with \(N\). Now, if \(S_1'/S_2>\log 2N\) there must be an interval \((n,n+L]\) containing two or more primes. The first paper uses a slightly different approach, but the second paper pursues the above line. Roughly speaking the terms in \(S_1'-S_1\) (that is to say, terms \(n+h\) where \(h\) is not one of the \(h_j\)) make a contribution of order \(L/(\log 2N)\) times \(S_1\). If we use only the Bombieri–Vinogradov value \(\theta=1/2\) then by taking \(k\) large enough we can make the limit in (6) as close to 1 as we like. Thus, if \(L=c\log 2N\) with any small \(c>0\), we may make \(S_1'/S_2>\log 2N\). The result (\(*\)) then follows. The second paper quantifies this approach. There are two major technical issues to be overcome. Firstly one must handle \(k\)-tuples uniformly for values \(k\) tending to infinity. Secondly, the error term \(x(\log x)^{-A}\) in the Bombieri–Vinogradov Theorem (2) needs to be improved. This is achieved by paying special attention to the possible effect of exceptional real zeros of Dirichlet \(L\)-functions. It is clear that one needs to work with \(k\)-tuples for which \(k\leq L\). A second fundamental constraint arises, namely that \(L\gg (\log 3N)/k\), and these limit the method to intervals of length \(\gg (\log 3N)^{1/2}\).
The methods in the second paper may be applied to primes from sets other than intervals. Thus it is shown that if \(\mathcal{A}\) is any set of natural numbers for which \[ \#\{a\in\mathcal{A}: a\leq N\}\gg (\log N)^{1/2}(\log\log N)^2 \] then infinitely many elements of \(\mathcal{A}-\mathcal{A}\) are differences of two primes.

11N05 Distribution of primes
11N36 Applications of sieve methods
11N13 Primes in congruence classes
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