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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.

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