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 $$A^D(N)= \{ Dq-2 : q\in P, N/2 < q \leq N \}$$ (sorry I can't get the curly brackets to work). 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
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 $$A^D(N)= { Dq-2 : q\in P, N/2 < q \leq N }$$ (sorry I can't get the curly brackets to work). 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