Question

I'm new to sieve theory, and I'm trying desperately to understand Selberg's sieve. I would like to apply the sieve to give me a nice upper bound on primes of the set $$A^D(N)= \{ Dq-2 : q\in P, N/2 < q \leq N \} $$ But basically, for a fixed N, I would like $A^D(N)$ to be the set of elements of the form $Dq-2$ for a fixed positive integer $D$ and letting $p$ run through all primes between $N/2$ and $N$. Now, as I said, I'm trying to apply Selberg's sieve, but as I don't really know what I'm doing, I'm a bit confused. Now, if I understand it correctly could I then say that $$S(A^D(N),N/2,N/2) \leq \frac{\pi(N)-\pi(N/2)}{L_p(z)}+O \Big( \frac{z^2}{L_p(x)^2} \Big)$$ Where $S(A^D(N),N/2,N/2)$ is the number of elements of $A^D(N)$ which are prime and $$L_p(z)=\sum_{n\leq z}^{n\vert P} \frac{\mu(n)^2}{\phi(n)}.$$ where $$\frac{1}{\phi(n)}=\frac{1}{n}\prod_{p\vert n} \frac{1}{1-1/p}.$$ I got this from a pater called "Sketch of the Selberg Sieve method" By Sean Prendville (January, 2008) where he describes the Selberg Sieve not on $A^D(N)$ but on the set of all integers between some positive integer $x$, and $x+y$. I'm sure some of it is wrong, or that I totally misinterpreted, but I would like to know if this is right, and if it is, where do I go from here? (especially with dealing with $L_p$). I appreciate any help, but please keep in mind that I'm sixteen years old and live in the Bronx. As dumbed down as possible would be greatly appreciated..this is all new to me. Much appreciated, Alexis D. Botros

Answer 1

Getting an upper bound here is, at the level of research, a simple exercise. To do it the quickest way with least background, I'd suggest using the large sieve. Here you are essentially looking at estimating the number of integers between $N/2$ and $N$ which, modulo primes $r\leq \sqrt{N}$, are neither $0$ nor $2\bar{D}$, where $\bar{D}$ is the inverse of $D$ modulo $r$ (and the second condition drops if $r$ divides $D$). The large sieve inequality tells you that this number is bounded by $N/J$ where $J=\sum_{n\leq \sqrt{N}}\mu(n)^2\prod_{r\mid n}{2/(r-2)}$, with $2/(r-2)$ replaced by $1/(r-1)$ when there is a single condition modulo $r$ ($r$ is again restricted to primes). Getting a lower bound for $J$ is again standard number theory, but not obvious of course at first sight. One gets $J\gg (\log N)^2$ (the key reason behind the exponent $2$ is that there are two excluded classes modulo each prime $r$), and therefore the number you want to estimate is $\ll N/(\log N)^2$. It is also fairly easy to obtain a result uniform in terms of $D$ along these lines.

For references, I suggest an old survey of H. Montgomery, "The analytic principle of the large sieve", which is available online (Bulletin of the AMS, 1984), and I think googling reveals also a very good short write-up by B. Green.