A first example of a sieve — I don't get the point
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The following is taken from the book Opera de Cribro. We take a finite sequence of non-negative real numbers $mathcal{A} = (a_n)$, $n leq x$, and a general set $mathcal{P}$ of primes. We write
$$P(z) = prod_{p in mathcal{P},,p < z} p$$
and our goal is to estimate the "sifting function"
$$S(mathcal{A},z) = sum_{n leq x,,(n,P(z)) = 1} a_n,.$$
The writers proceed to give an example. They consider $mathcal{P} = {p : p notequiv3 text{ mod $4$}}$ and $mathcal{A} = {m^2 + 1 leq x}$. This definition of $mathcal{A}$ already confusing to me, but I presume this means
$$a_n = begin{cases}
1 qquad text{if $n = m^2 + 1 leq x$}; \
0 qquad text{otherwise.}end{cases}$$
They then drop the following bomb.
Were we able to get a positive lower bound for $S(mathcal{A},sqrt{x})$ we would be producing primes of the form $m^2 + 1$.
I don't see at all how this is true. What am I missing?
number-theory elementary-number-theory analytic-number-theory sieve-theory
asked Nov 21 at 12:04
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The following is taken from the book Opera de Cribro. We take a finite sequence of non-negative real numbers $mathcal{A} = (a_n)$, $n leq x$, and a general set $mathcal{P}$ of primes. We write
$$P(z) = prod_{p in mathcal{P},,p < z} p$$
and our goal is to estimate the "sifting function"
$$S(mathcal{A},z) = sum_{n leq x,,(n,P(z)) = 1} a_n,.$$
The writers proceed to give an example. They consider $mathcal{P} = {p : p notequiv3 text{ mod $4$}}$ and $mathcal{A} = {m^2 + 1 leq x}$. This definition of $mathcal{A}$ already confusing to me, but I presume this means
$$a_n = begin{cases}
1 qquad text{if $n = m^2 + 1 leq x$}; \
0 qquad text{otherwise.}end{cases}$$
They then drop the following bomb.
Were we able to get a positive lower bound for $S(mathcal{A},sqrt{x})$ we would be producing primes of the form $m^2 + 1$.
I don't see at all how this is true. What am I missing?
number-theory elementary-number-theory analytic-number-theory sieve-theory
asked Nov 21 at 12:04
guest
834
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The following is taken from the book Opera de Cribro. We take a finite sequence of non-negative real numbers $mathcal{A} = (a_n)$, $n leq x$, and a general set $mathcal{P}$ of primes. We write
$$P(z) = prod_{p in mathcal{P},,p < z} p$$
and our goal is to estimate the "sifting function"
$$S(mathcal{A},z) = sum_{n leq x,,(n,P(z)) = 1} a_n,.$$
The writers proceed to give an example. They consider $mathcal{P} = {p : p notequiv3 text{ mod $4$}}$ and $mathcal{A} = {m^2 + 1 leq x}$. This definition of $mathcal{A}$ already confusing to me, but I presume this means
$$a_n = begin{cases}
1 qquad text{if $n = m^2 + 1 leq x$}; \
0 qquad text{otherwise.}end{cases}$$
They then drop the following bomb.
Were we able to get a positive lower bound for $S(mathcal{A},sqrt{x})$ we would be producing primes of the form $m^2 + 1$.
I don't see at all how this is true. What am I missing?
number-theory elementary-number-theory analytic-number-theory sieve-theory
asked Nov 21 at 12:04
guest
834
The following is taken from the book Opera de Cribro. We take a finite sequence of non-negative real numbers $mathcal{A} = (a_n)$, $n leq x$, and a general set $mathcal{P}$ of primes. We write
$$P(z) = prod_{p in mathcal{P},,p < z} p$$
and our goal is to estimate the "sifting function"
$$S(mathcal{A},z) = sum_{n leq x,,(n,P(z)) = 1} a_n,.$$
The writers proceed to give an example. They consider $mathcal{P} = {p : p notequiv3 text{ mod $4$}}$ and $mathcal{A} = {m^2 + 1 leq x}$. This definition of $mathcal{A}$ already confusing to me, but I presume this means
$$a_n = begin{cases}
1 qquad text{if $n = m^2 + 1 leq x$}; \
0 qquad text{otherwise.}end{cases}$$
They then drop the following bomb.
Were we able to get a positive lower bound for $S(mathcal{A},sqrt{x})$ we would be producing primes of the form $m^2 + 1$.
I don't see at all how this is true. What am I missing?
number-theory elementary-number-theory analytic-number-theory sieve-theory
number-theory elementary-number-theory analytic-number-theory sieve-theory
asked Nov 21 at 12:04
guest
834
asked Nov 21 at 12:04
guest
834
asked Nov 21 at 12:04
guest
834
asked Nov 21 at 12:04
guest
834
asked Nov 21 at 12:04
guest
834
834
add a comment |
add a comment |
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