# The Green-Tao theorem and positive binary quadratic forms

Some time ago I asked a question on consecutive numbers represented integrally by an integral positive binary quadratic form. It has occurred to me that, instead, the Green-Tao theorem may include a result on arithmetic progressions represented by a positive binary form. So my question is whether that is the case, do we already know that a positive binary form represents arbitrarily long arithmetic progressions? These would be primes in this general setting, thus quite different from consecutive integers of course.

My main reference is
David A. Cox, Primes of the form $x^2 + n y^2.$ He defines the Dirichlet density on page 169. Then he states the Chebotarev Density Theorem (8.17) on page 170. Finally he gives the Dirichlet density of primes represented by a positive binary form on page 188, Theorem 9.12. EDIT::: Not difficult to state: with discriminant $\Delta < 0$ and class number $h(\Delta),$ if the form is ambiguous (such as the principal form) the Dirichlet density of the set of primes it represents is $$\frac{1}{2 h(\Delta)},$$ while if the form is not ambiguous the Dirichlet density is $$\frac{1}{h(\Delta)}.$$ On page 190 he does the example $\Delta = -56.$ Here $x^2 + 14 y^2$ represents a set of primes with Dirichlet density $1/8,$ while $2 x^2 + 7 y^2$ also gets density $1/8,$ but in the other genus $3 x^2 + 2 x y + 5 y^2$ and $3 x^2 - 2 x y + 5 y^2$ each represent the same set of primes with density $1/4.$ Note on page 195 we have Exercise 9.17, that the sum of these densities for any discriminant must be $1/2.$ A little fiddling, not mentioned in the book, shows that each genus (of a fixed discriminant $\Delta$) represents the same total density, something we really want because of the relationship between genera and arithmetic progressions of primes.

From an earlier answer by David Hansen it would appear that the only missing ingredient is a comparison between Dirichlet density and "relative density." I have never been sure on this point, does a positive binary form represent the same relative density of primes as its Dirichlet density? Anyway, see: Is the Green-Tao theorem true for primes within a given arithmetic progression?

My earlier question, about which I should say that I have come to believe there is no upper bound on the length of intervals represented, despite the great difficulty finding examples: Can a positive binary quadratic form represent 14 consecutive numbers?

On the Green-Tao theorem itself:

same-day EDIT: I looked up Dirichlet density on wikipedia. As relates to the earlier David Hansen answer I liked, wikipedia stops short of saying that the Dirichlet density of primes in an arithmetic progression is identical to the relative density. Personally, I cannot see how the relative density could be anything else, but that is just my opinion. So I think I am also asking for references that prove the relative density is equal to the Dirichlet density in some naturally-occurring situations.

http://en.wikipedia.org/wiki/Dirichlet_density

Later on the same day EDIT: There is stronger language in this next wikipedia page, so I think we can conclude that for primes in an arithmetic progression the Dirichlet density and the relative density are equal, but I would still like a more substantial reference. I'm the nervous type. I worry.

-

Dear Kevin, Jurgen Neukirch "Class Field Theory" points to Serre "A Course in Arithmetic," and on page 76 he says that the set of primes $p$ such that a fixed polynomial has a root $\pmod p$ has a natural density, and refers to K. Prachar "Primzahlverteilung" chapter 5 section 7. By results in the Cox book, this means that the principal form $x^2 + n y^2$ or $x^2 + x y + k y^2$ does represent a set of primes with a natural density, therefore equalling the Dirichlet density. And by the result on arithmetic progressions, a full genus of forms has a natural density. Getting there. – Will Jagy Jun 15 '10 at 16:33
The text passage is a discussion about the representation of primes by positive binary quadratic forms and the relationship between Dirichlet density and relative density. It also mentions the Green-Tao theorem and its potential implications for arithmetic progressions represented by such forms.
no_question

the passage does not contain a direct mathematical question or calculation that requires an answer. it is more of a discussion or exploration of a mathematical concept.
Given a text passage, determine:
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First, describe the content structure you found in 1-2 sentences.

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# The Green-Tao theorem and positive binary quadratic forms

Some time ago I asked a question on consecutive numbers represented integrally by an integral positive binary quadratic form. It has occurred to me that, instead, the Green-Tao theorem may include a result on arithmetic progressions represented by a positive binary form. So my question is whether that is the case, do we already know that a positive binary form represents arbitrarily long arithmetic progressions? These would be primes in this general setting, thus quite different from consecutive integers of course.

My main reference is
David A. Cox, Primes of the form $x^2 + n y^2.$ He defines the Dirichlet density on page 169. Then he states the Chebotarev Density Theorem (8.17) on page 170. Finally he gives the Dirichlet density of primes represented by a positive binary form on page 188, Theorem 9.12. EDIT::: Not difficult to state: with discriminant $\Delta < 0$ and class number $h(\Delta),$ if the form is ambiguous (such as the principal form) the Dirichlet density of the set of primes it represents is $$\frac{1}{2 h(\Delta)},$$ while if the form is not ambiguous the Dirichlet density is $$\frac{1}{h(\Delta)}.$$ On page 190 he does the example $\Delta = -56.$ Here $x^2 + 14 y^2$ represents a set of primes with Dirichlet density $1/8,$ while $2 x^2 + 7 y^2$ also gets density $1/8,$ but in the other genus $3 x^2 + 2 x y + 5 y^2$ and $3 x^2 - 2 x y + 5 y^2$ each represent the same set of primes with density $1/4.$ Note on page 195 we have Exercise 9.17, that the sum of these densities for any discriminant must be $1/2.$ A little fiddling, not mentioned in the book, shows that each genus (of a fixed discriminant $\Delta$) represents the same total density, something we really want because of the relationship between genera and arithmetic progressions of primes.

From an earlier answer by David Hansen it would appear that the only missing ingredient is a comparison between Dirichlet density and "relative density." I have never been sure on this point, does a positive binary form represent the same relative density of primes as its Dirichlet density? Anyway, see: Is the Green-Tao theorem true for primes within a given arithmetic progression?

My earlier question, about which I should say that I have come to believe there is no upper bound on the length of intervals represented, despite the great difficulty finding examples: Can a positive binary quadratic form represent 14 consecutive numbers?

On the Green-Tao theorem itself:

same-day EDIT: I looked up Dirichlet density on wikipedia. As relates to the earlier David Hansen answer I liked, wikipedia stops short of saying that the Dirichlet density of primes in an arithmetic progression is identical to the relative density. Personally, I cannot see how the relative density could be anything else, but that is just my opinion. So I think I am also asking for references that prove the relative density is equal to the Dirichlet density in some naturally-occurring situations.

http://en.wikipedia.org/wiki/Dirichlet_density

Later on the same day EDIT: There is stronger language in this next wikipedia page, so I think we can conclude that for primes in an arithmetic progression the Dirichlet density and the relative density are equal, but I would still like a more substantial reference. I'm the nervous type. I worry.

-

Dear Kevin, Jurgen Neukirch "Class Field Theory" points to Serre "A Course in Arithmetic," and on page 76 he says that the set of primes $p$ such that a fixed polynomial has a root $\pmod p$ has a natural density, and refers to K. Prachar "Primzahlverteilung" chapter 5 section 7. By results in the Cox book, this means that the principal form $x^2 + n y^2$ or $x^2 + x y + k y^2$ does represent a set of primes with a natural density, therefore equalling the Dirichlet density. And by the result on arithmetic progressions, a full genus of forms has a natural density. Getting there. – Will Jagy Jun 15 '10 at 16:33
user
The text passage is a discussion about the representation of primes by positive binary quadratic forms and the relationship between Dirichlet density and relative density. It also mentions the Green-Tao theorem and its potential implications for arithmetic progressions represented by such forms.

TYPE: NO_QUESTION

The passage does not contain a direct mathematical question or calculation that requires an answer. It is more of a discussion or exploration of a mathematical concept.