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edited Feb 17 2011 at 3:40 |
I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. Its apparent improvement over previous works is that it covers not just quadratic forms plus a constant but quadratic forms plus linear forms plus a constant.
I would like to know what the current state of knowledge is for this sort of problem.
- This paper covers only the case of polynomials that "depend essentially on two variables". This has surely not been improved, else the H-L Conjecture E, F, etc. would have been solved.
- The paper allows only integers, not half-integers, as coefficients, and so does not give the number of primes, e.g., in A117112.
- It does not give the constant (or prove that a constant exists!) for the densities it finds.
The third is my main interest at this point. References would be appreciated, whether to research papers, survey papers, or standard texts. I also have some interest in the cubic case, if anything is known. It seems that even characterizing which two-variable cubics represent an infinite number of primes is open...? Certainly the conditions Iwaniec gives for two-variable quadratics do not suffice to give an infinity of primes.
For example, with the polynomial $P(x,y)=ax^2+bxy+cy^2+ex+fy+g$, setting $\Delta=b^2-4ac$ and $D=af^2-bef+ce^2+g\Delta$ and $C(x)=\sum_{p=P(x,y)\le x}$1, [1] shows that
$$\frac{x}{(\log x)^{1.5}}\ll C(x)\ll\frac{x}{(\log x)^{1.5}}$$
for all $P$ with $D\neq0$ and $\Delta$ not a square, but is it known that $C(x)\sim kx/(\log x)^{1.5}$ for some $k$? Are values of $e$ or $E$ known such that
$$e<\liminf\frac{C(x)(\log x)^{1.5}}{x}$$
or
$$\limsup\frac{C(x)(\log x)^{1.5}}{x}< E$$
?
similarly
Similarly, if $D=0$ or $\Delta$ is a square do we know when
$$\ell=\lim\frac{C(x)\log x}{x}$$
exists and what its value is?
Related question: Who should I cite for these results? I'm never sure of when there might have been simultaneous discoveries or other reasons for priority issues.
[1] Iwaniec, H. (1974). Primes represented by quadratic polynomials in two variables. Acta Arithmetica 24, pp. 435–459.
[2] Hardy, G. H., & Littlewood, J. E. (1923). Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta Mathematica 44:1, pp. 1-70.
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edited Feb 16 2011 at 21:54 |
I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. Its apparent improvement over previous works is that it covers not just quadratic forms plus a constant but quadratic forms plus linear forms plus a constant.
I would like to know what the current state of knowledge is for this sort of problem.
- This paper covers only the case of polynomials that "depend essentially on two variables". This has surely not been improved, else the H-L Conjecture E, F, etc. would have been solved.
- The paper allows only integers, not half-integers, as coefficients, and so does not give the number of primes, e.g., in A117112.
- It does not give the constant (or prove that a constant exists!) for the densities it finds.
The third is my main interest at this point. References would be appreciated, whether to research papers, survey papers, or standard texts. I also have some interest in the cubic case, if anything is known. It seems that even characterizing which two-variable cubics represent an infinite number of primes is open...? Certainly the conditions Iwaniec gives for two-variable quadratics do not suffice to give an infinity of primes.
For example, with the polynomial $P(x,y)=ax^2+bxy+cy^2+ex+fy+g$, setting $\Delta=b^2-4ac$ and $D=af^2-bef+ce^2+g\Delta$ and $C(x)=\sum_{p=P(x,y)\le x}$1, [1] shows that
$$\frac{x}{(\log x)^{1.5}}\ll C(x)\ll\frac{x}{(\log x)^{1.5}}$$
for all $P$ with $D\neq0$ and $\Delta$ not a square, but is it known that $C(x)\sim kx/(\log x)^{1.5}$ for some $k$? Are values of $e$ or $E$ known such that
$$e<\liminf\frac{C(x)(\log x)^{1.5}}{x}$$
or
$$\limsup\frac{C(x)(\log x)^{1.5}}{x}< E$$
?
similarly, if $D=0$ or $\Delta$ is a square do we know when
$$\ell=\lim\frac{C(x)\log x}{x}$$
exists and what its value is?
Related question: Who should I cite for these results? I'm never sure of when there might have been simultaneous discoveries or other reasons for priority issues.
[1] Iwaniec, H. (1974). Primes represented by quadratic polynomials in two variables. Acta Arithmetica 24, pp. 435–459.
[2] Hardy, G. H., & Littlewood, J. E. (1923). Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta Mathematica 44:1, pp. 1-70.
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edited Feb 15 2011 at 22:03 |
I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. Its apparent improvement over previous works is that it covers not just quadratic forms plus a constant but quadratic forms plus linear forms plus a constant.
I would like to know what the current state of knowledge is for this sort of problem.
- This paper covers only the case of polynomials that "depend essentially on two variables". This has surely not been improved, else the H-L Conjecture E, F, etc. would have been solved.
- The paper allows only integers, not half-integers, as coefficients, and so does not give the number of primes, e.g., in A117112.
- It does not give the constant (or prove that a constant exists!) for the densities it finds.
The third is my main interest at this point. References would be appreciated, whether to research papers, survey papers, or standard texts. I also have some interest in the cubic case, if anything is known.
For example, with the polynomial $P(x,y)=ax^2+bxy+cy^2+ex+fy+g$, setting $\Delta=b^2-4ac$ and $D=af^2-bef+ce^2+g\Delta$ and $C(x)=\sum_{p=P(x,y)\le x}$1, [1] shows that
$$\frac{x}{(\log x)^{1.5}}\ll C(x)\ll\frac{x}{(\log x)^{1.5}}$$
for all $P$ with $D\neq0$ and $\Delta$ not a square, but is it known that $C(x)\sim kx/(\log x)^{1.5}$ for some $k$? Are values of $e$ or $E$ known such that
$$e<\liminf\frac{C(x)(\log x)^{1.5}}{x}$$
or
$$\limsup\frac{C(x)(\log x)^{1.5}}{x}< E$$
?
similarly, if $D=0$ or $\Delta$ is a square do we know when
$$e<\lim\frac{C(x)\log $\ell=\lim\frac{C(x)\log x}{x}$$
exists and what its value is?
Related question: Who should I cite for these results? I'm never sure of when there might have been simultaneous discoveries or other reasons for priority issues.
[1] Iwaniec, H. (1974). Primes represented by quadratic polynomials in two variables. Acta Arithmetica 24, pp. 435–459.
[2] Hardy, G. H., & Littlewood, J. E. (1923). Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta Mathematica 44:1, pp. 1-70.
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edited Feb 15 2011 at 3:02 |
I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. Its apparent improvement over previous works is that it covers not just quadratic forms plus a constant but quadratic forms plus linear forms plus a constant.
I would like to know what the current state of knowledge is for this sort of problem.
- This paper covers only the case of polynomials that "depend essentially on two variables". This has surely not been improved, else the H-L Conjecture E, F, etc. would have been solved.
- The paper allows only integers, not half-integers, as coefficients, and so does not give the number of primes, e.g., in A117112.
- It does not give the constant (or prove that a constant exists!) for the densities it finds.
The third is my main interest at this point. References would be appreciated, whether to research papers, survey papers, or standard texts. I also have some interest in the cubic case, if anything is known.
For example, with the polynomial $P(x,y)=ax^2+bxy+cy^2+ex+fy+g$, setting $\Delta=b^2-4ac$ and $D=af^2-bef+ce^2+g\Delta$ and $C(x)=\sum_{p=P(x,y)\le x}$1, [1] shows that
$$\frac{x}{(\log x)^{1.5}}\ll C(x)\ll\frac{x}{(\log x)^{1.5}}$$
for all $P$ with $D\neq0$ and $\Delta$ not a square, but is it known that $C(x)\sim kx/(\log x)^{1.5}$ for some $k$? Are values of $e$ or $E$ known such that
$$e<\liminf\frac{C(x)(\log x)^{1.5}}{x}$$
or
$$\limsup\frac{C(x)(\log x)^{1.5}}{x}< E$$
?
similarly, if $D=0$ or $\Delta$ is a square do we know $e$ or $E<1$ with
when
$$e<\liminf\frac{C(x)\log <\lim\frac{C(x)\log x}{x}$$
or
$$\limsup\frac{C(x)\log x}{x}< E$$
?
Of course $E=1$ exists and what its value isobvious in this case.?
Related question: Who should I cite for these results? I'm never sure of when there might have been simultaneous discoveries or other reasons for priority issues.
[1] Iwaniec, H. (1974). Primes represented by quadratic polynomials in two variables. Acta Arithmetica 24, pp. 435–459.
[2] Hardy, G. H., & Littlewood, J. E. (1923). Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta Mathematica 44:1, pp. 1-70.
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edited Feb 15 2011 at 2:21 |
I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. Its apparent improvement over previous works is that it covers not just quadratic forms plus a constant but quadratic forms plus linear forms plus a constant.
I would like to know what the current state of knowledge is for this sort of problem.
- This paper covers only the case of polynomials that "depend essentially on two variables". This has surely not been improved, else the H-L Conjecture E, F, etc. would have been solved.
- The paper allows only integers, not half-integers, as coefficients, and so does not give the number of primes, e.g., in A117112.
- It does not give the constant (or prove that a constant exists!) for the densities it finds.
The third is my main interest at this point. References would be appreciated, whether to research papers, survey papers, or standard texts. I also have some interest in the cubic case, if anything is known.
For example, with the polynomial $P(x,y)=ax^2+bxy+cy^2+ex+fy+g$, setting $\Delta=b^2-4ac$ and $D=af^2-bef+ce^2+\Delta$ D=af^2-bef+ce^2+g\Delta$ and $C(X)=\sum_{p=P(x,y)\le C(x)=\sum_{p=P(x,y)\le x}$1, [1] shows that
$$\frac{x}{(\log x)^{1.5}}\ll C(x)\ll\frac{x}{(\log x)^{1.5}}$$
for all $P$ with $D\neq0$ and $\Delta$ not a square, but is it known that $C(x)\sim kx/(\log x)^{1.5}$ for some $k$? Are values of $e$ or $E$ known such that
$$e<\liminf\frac{C(x)(\log x)^{1.5}}{x}$$
or
$$\limsup\frac{C(x)(\log x)^{1.5}}{x}< E$$
?
similarly, if $D=0$ or $\Delta$ is a square do we know $e$ or $E<1$ with
$$e<\liminf\frac{C(x)\log x}{x}$$
or
$$\limsup\frac{C(x)\log x}{x}< E$$
?
Of course $E=1$ is obvious in this case.
Related question: Who should I cite for these results? I'm never sure of when there might have been simultaneous discoveries or other reasons for priority issues.
[1] Iwaniec, H. (1974). Primes represented by quadratic polynomials in two variables. Acta Arithmetica 24, pp. 435–459.
[2] Hardy, G. H., & Littlewood, J. E. (1923). Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta Mathematica 44:1, pp. 1-70.
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edited Feb 14 2011 at 20:45 |
I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. Its apparent improvement over previous works is that it covers not just quadratic forms plus a constant but quadratic forms plus linear forms plus a constant.
I would like to know what the current state of knowledge is for this sort of problem.
- This paper covers only the case of polynomials that "depend essentially on two variables". This has surely not been improved, else the H-L Conjecture E, F, etc. would have been solved.
- The paper allows only integers, not half-integers, as coefficients, and so does not give the number of primes, e.g., in A117112.
- It does not give the constant (or prove that a constant exists!) for the densities it finds.
The third is my main interest at this point. References would be appreciated, whether to research papers, survey papers, or standard texts. I also have some interest in the cubic case, if anything is known.
For example, with the polynomial $P(x,y)=ax^2+bxy+cy^2+ex+fy+g$, setting $\Delta=b^2-4ac$ and $D=af^2-bef+ce^2+\Delta$ and $C(X)=\sum_{p=P(x,y)\le x}$1, [1] shows that
$$\frac{x}{(\log x)^{1.5}}\ll C(x)\ll\frac{x}{(\log x)^{1.5}}$$
but is it known that $C(x)\sim kx/(\log x)^{1.5}$ for some $k$? Are values of $e$ or $E$ known such that
$$e<\liminf\frac{C(x)(\log x)^{1.5}}{x}$$
or
$$\limsup\frac{C(x)(\log x)^{1.5}}{x}< E$$
?
Related question: Who should I cite for these results? I'm never sure of when there might have been simultaneous discoveries or other reasons for priority issues.
[1] Iwaniec, H. (1974). Primes represented by quadratic polynomials in two variables. Acta Arithmetica 24, pp. 435–459.
[2] Hardy, G. H., & Littlewood, J. E. (1923). Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta Mathematica 44:1, pp. 1-70.
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asked Feb 14 2011 at 6:29 |
Primes represented by two-variable quadratic polynomials
I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. Its apparent improvement over previous works is that it covers not just quadratic forms plus a constant but quadratic forms plus linear forms plus a constant.
I would like to know what the current state of knowledge is for this sort of problem.
- This paper covers only the case of polynomials that "depend essentially on two variables". This has surely not been improved, else the H-L Conjecture E, F, etc. would have been solved.
- The paper allows only integers, not half-integers, as coefficients, and so does not give the number of primes, e.g., in A117112.
- It does not give the constant (or prove that a constant exists!) for the densities it finds.
The third is my main interest at this point. References would be appreciated, whether to research papers, survey papers, or standard texts. I also have some interest in the cubic case, if anything is known.
Related question: Who should I cite for these results? I'm never sure of when there might have been simultaneous discoveries or other reasons for priority issues.
[1] Iwaniec, H. (1974). Primes represented by quadratic polynomials in two variables. Acta Arithmetica 24, pp. 435–459.
[2] Hardy, G. H., & Littlewood, J. E. (1923). Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta Mathematica 44:1, pp. 1-70.
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