The least prime number represented by a binary quadratic form

The least prime number represented by a binary quadratic form

Naser T. Sardari
July 16, 2019
Abstract.

Let be a fundamental discriminant and be the class number of the imaginary quadratic field . Moreover, assume that is the number of the split primes with norm less than in and is the number of the classes of the binary quadratic forms of discriminant which represents a prime number less than . We prove that

where is the number of the primes less than and the implicit constant in is independent of . As a result, by assuming the Riemann hypothesis for the Dirichlet L-function , at least number of the ideal classes of contain a prime ideal with a norm less than the optimal bound where is an absolute positive constant independent of . More generally, let be a bounded degree number field over with the discriminant and the class number We conjecture that a positive proportion of the ideal classes of contain a prime ideal with a norm less than .

1. Introduction

1.1. Motivation

In this paper, we consider the problem of giving the optimal upper bound on the least prime number represented by a binary quadratic form in terms of its discriminant. Giving sharp upper bound on the least prime number represented by a binary quadratic form is crucial in the analysis of the complexity of some algorithms in quantum compiling. In particular, Ross and Selinger’s algorithm for the optimal navigation of -axis rotations in by quantum gates [RS14] and its -adic analogue for finding the shortest path between two diagonal vertices of LPS Ramanujan graphs [Sar17]. In [Sar17], we proved that these heuristic algorithms run in polynomial time under a Cramér type conjecture on the distribution of the inverse image of the integers representable as a sum of two squares by a binary quadratic from; see [Sar17, Conjecture 1.4]. In this paper we show that this Cramér type conjecture holds with a positive probability that depends only on . In [Sar18], we proved that for a given fundamental discriminant , by assuming the generalized Riemann hypothesis for the zeta function of the Hilbert class field of the imaginary quadratic field , of the binary quadratic forms of discriminant represent a prime number less than as . In this paper, we remove the GRH assumption and show that unconditionally with probability at least a binary quadratic forms of discriminant represent a prime number smaller than any fixed scalar multiple of , where is an absolute constant independent of . As a result, we prove that if for some then a positive proportion of the binary quadratic forms of discriminant represent a prime number smaller than any fixed scalar multiple of More precisely, we prove the following result about the least prime represented by a binary quadratic form of fixed discriminant .

Theorem 1.1.

Assume that is a fundamental discriminant, and are the class number and the number of primes in the interval that splits in the imaginary quadratic field . Moreover, Let be the number of the classes of the binary quadratic forms of discriminant which represents a prime number less than . Then

where is the number of the primes less than and the implicit constant in is independent of .

Remark 1.2.

Note that by Chebotarev’s density theorem or Dirichlet’s theorem we have as By assuming Riemann hypothesis or even zero-free region of width for the Dirichlet L-function , we have for any where Since then under GRH we have for any and it follows that the above proposed algorithms give a probabilistic polynomial time algorithm for navigating and .

Next, we show that our result is optimal up to a scalar. Namely, if a positive proportion of the binary quadratic forms of discriminant , represent a prime number less than then we have the following lower bound on

We give a proof of this claim in what follows. Let be the genus class of the binary quadratic form of discriminant and denote the sum of the representation of by all the classes of binary quadratic forms of discriminant

By the classical formula due to Dirichlet we have

(1.1)

where,

This means that the multiplicity of representing a prime number by all the binary quadratic forms of a fixed negative discriminant is bounded by

(1.2)

Assume that a positive proportion of binary quadratic forms represent a prime number smaller than . Let denote the number of the pairs such that is a prime number represented by . We proceed by giving a double counting formula for . By our assumption a positive proportion of binary quadratic forms of discriminant represent a prime number less than , then

(1.3)

On the other hand,

By inequality 1.2,

where is the number of primes that splits in By the above inequality and inequality 1.3, we obtain

By Siegel’s lower bound , it follows that

This completes the proof of our claim.

1.2. The generalized Minkowski’s bound for the prime ideals

It follows from our result that a positive proportion of the ideal classes of contains a prime ideal with a norm less than the optimal bound More precisely, let be a fundamental discriminant, which means is square-free and . Let denote the ideal class group of and be the norm of the imaginary quadratic field Given an integral ideal , let be the following class of the integral binary quadratic form defined up to the action of

(1.4)

where , and identifies the integral ideal with . It follows that only depends on the ideal class This gives an isomorphism between and the orbits of the integral binary quadratic forms of discriminant under the action of . Note that if represents the prime number then for some . Then, the principal ideal factors into the product of and where and belongs to the inverse of the ideal class . Let denote the number of the ideal classes of the ideal class group of that contains a prime ideal with norm less than . Hence, we have the following corollary from Theorem 1.1.

Corollary 1.3.

We have

where the implicit constant in is independent of .

More generally, let be a number field of bounded degree over with the discriminant and the class number . Then we have the following conjecture which generalizes Minkowski’s bound for the prime ideals.

Conjecture 1.4.

Let be a number field of bounded degree over with the discriminant and the class number . Then a positive proportion (only depends on ) of the ideal classes in the ideal class group of contain a prime ideal with a norm less than any fixed scalar multiple of .

Next, we show that these bounds are compatible with the random model for the prime numbers known as Cramér’s model. We cite the following formulation of the Cramér model from [Sou07].

Cramér’s model 1.5.

The primes behave like independent random variables with (the number is ‘prime’) with probability , and (the number is ‘composite’) with probability

Note that each class of the integral binary quadratic forms is associated to a Heegner point in . By the equdistribution of Heegner points in , it follows that almost all classes of the integral quadratic forms has a representative such that the coefficients of are bounded by any function growing faster than :

for any function defined on integers such that as . We show this claim in what follows. We consider the set of representative of the Heegner points inside the Gauss fundamental domain of and denote them by for . They are associated to the roots of a representative of a binary quadratic form in the ideal class group. By the equidistribution of Heegner points in and the fact that the volume of the Gauss fundamental domain decay with rate near the cusp, it follows that for almost all if is the Heegner point inside the Gauss fundamental domain associated to then

(1.5)

where is any function such that as Let be the quadratic forms associated to that has as its root. Then

where and . By inequality (1.5), we have

(1.6)

By the above inequalities and , it follows that

(1.7)

This concludes our claim. Next, we give a heuristic upper bound on the size of the smallest prime number represented by a binary quadratic forms of discriminant that satisfies (1.7). Since is square-free, there is no local restriction for representing prime numbers. So, by the Cramér’s model and consideration of the Hardy-Littlewood local measures, we expect that for a positive proportion of the classes of the binary quadratic forms there exists an integral point such that and is a prime number. We have

(1.8)

We may take to be any constant in the above estimate. Therefore, we expect that a positive proportion of the quadratic forms of discriminant represent a prime number less than By a similar analysis, we expect that almost all binary quadratic forms of discriminant represent a prime number less than In other words, almost all ideal classes of contain a prime ideal with norm less than In [Sar18], we proved this result by assuming the generalized Riemann hypothesis for the zeta function of the Hilbert class field of the imaginary quadratic field We conjectured that this type of generalized Minkowsky’s bounds holds for every number fields.

Conjecture 1.6.

Let be a number field of the bounded degree over with the discriminant and the class number . Then almost all ideal classes in the ideal class group of contain a prime ideal with norm less than for some . Note that by the Brauer-Siegel Theorem and GRH, we have

1.3. Repulsion of the prime ideals near the cusp

As we noted above, based on the Cramér’s model we expect that the split prime numbers randomly distributed among the ideal classes of , and hence with a positive probability that is independent of , a quadratic form of discriminant represent a prime number less than a fixed scalar multiple of We may hope that every ideal class contain a prime ideal of size . Note that Cramér conjecture states that every short interval of size contains a prime number. By Linnik’s conjecture, every congruence class modulo contains a prime number less than This shows that small prime numbers covers all the short interval and congruence classes. However, we note that the family of binary quadratic forms of discriminant is different from the family of short intervals and its -adic analogue. Small primes are not covering all the class of binary quadratic forms. For example, the principal ideal class that is associated to the binary quadratic form repels prime number which means the least prime number represented by this form is bigger than compared to that is the upper bound for almost all binary quadratic forms under GRH. This feature is different from the analogues conjectures for the size of the least prime number in a given congruence classes modulo an integer (Linnik’s conjecture) and the distribution of prime numbers in short intervals (Cramér’s conjecture). We call this new feature the repulsion of small primes by the cusp. In fact, the binary quadratic forms with the associated Heegner point near the cusp repels prime numbers. This can be seen in equation (1.8), where could be as large as near the cusp but for a typical binary quadratic form it is bounded by This shows that the bound in the Conjecture 1.6 does not hold for every ideal class.

1.4. Method of the proof

Our strategy of the proof is based on our recent work on the distribution of the prime numbers in the short intervals. In [Sar], we proved that a positive proportion of the intervals of any fixed scalar multiple of in the dyadic interval contain a prime number. We also showed that a positive proportion of the congruence classes modulo contain a prime number smaller than any fixed scalar multiple of These result are compatible with Cramér’s Model.

We briefly describe our method here. We proceed by introducing some new notations and follow the previous ones. Let be a positive smooth weight function that is supported on and . Let that is derived from by scaling with . Let denote the number of the classes of binary quadratic forms of discriminant that represents a prime number inside the dyadic interval . Let denote the number of primes weighted by that are representable by the binary quadratic form . By the Cauchy-Schwarz inequality, we obtain

(1.9)

By Dirichlet formula in (1.1), is the weighted number of prime numbers inside the interval that splits in the quadratic field . So, we have

(1.10)

Next, we give a double counting formula for the sum . This sum counts pairs of primes weighted by such that and are represented by the same binary quadratic form class . Assume that is a representative of that class that represents two prime numbers and . With out loss of generality we assume that for some integers and such that

(1.11)

Since by the action of on the space of the integral binary quadratic forms we can find a representative of with the above form. Since represents then

for some integers and . We multiply both side of the above identity by and obtain

We use identity (1.11), and substitute in the above identity and obtain

Hence,

We change the variables to , and obtain

(1.12)

On the other hand if is a solution to the equation (1.12) for prime numbers and , then and are represented by the same binary quadratic form class in . Heuristically, this number is about , that is the number of distinct pairs of split primes inside the interval divided by the number of the classes of binary quadratic forms of discriminant plus the contribution of diagonal terms where . Therefore, we expect

(1.13)

In fact, by applying the Selberg upper bound sieve on the number of the prime solutions to the equation (1.12), we show that

(1.14)

Therefore, by the inequality (1.9), equation (1.10) and the above inequality, it follows that

This gives a proof of Theorem 1.1. Next, we briefly explain how we prove inequality (1.14). We begin by counting the number of the solutions to the equation (1.12) weighted by the smooth weight function where . We call them by the diagonal solutions. If then

Hence, for some prime number and . Therefore, the number of diagonal solutions to the equation (1.12) is the number of prime numbers weighted by that is

Next, we give an upper bound on the number of non-diagonal terms weighted by . Since and only if then

(1.15)

We fix and apply the Selberg upper bound sieve for giving a sharp upper bound up to a constant on the number of the prime solutions to the following equation weighted by .

(1.16)

More precisely, we give an upper bound on the weighted number of integral points lying on the following ternary quadric

(1.17)

where and do not have any prime divisor smaller than where and for a small power ; e.g. . In what follows, we explain the Selberg upper sieve. Assume that and are square-free integers. Let denote the number of the integral solutions weighted by to the equation

where and . Similarly, let be the same number where . We write for where It follows from the inclusion exclusion principal that; see [BF94, Lemma 8, Page 79]

(1.18)

Let denote the weighted number of the integral solutions to the equation (1.17) where and don’t have any prime divisor smaller than and denote the indicator function of the integers with no prime divisor less than . Since and are real numbers then we have the following upper bound on

(1.19)

Hence,

(1.20)

where

(1.21)

In Theorem 3.10, we give an asymptotic formula for with a power saving error term if This theorem is a quantitative version of Duke’s theorem on the equadistribution of the Heegner points. The proof of this theorem is the main technical part of our work. We apply the Siegel Mass formula on the ternary quadratic form in order to give the main term of as the product of the Hardy-Littlewood local densities. For giving a power saving upper bound on the error term we use Duke’s non-trivial bounds on the Fourier coefficients of weight Maass forms and our bound on the norm of the theta lift of weight Maass forms. We give the outline of the proof of Theorem 3.10 in the next section. By assuming this results the main term of the weighted number of integral points comes from the product of the local densities with a power saving error term Er

(1.22)

where and is given by

(1.23)

We explicitly compute these local densities in term of the quadratic character and as a result we have an explicit formula for the sieve densities where

(1.24)

For a squarefree integer , define

(1.25)

and let

(1.26)

where the sum is over squarefree variables By the fundamental theorem for Selberg sieve [FI10, Theorem 7.1], we have

In Lemma 2.7, we show that

Finally, by summing over and proving the analogue of Gallagher’s result on the average size of the Hardy-Littlewood singular series [Gal76, equation (3)], we prove inequality (1.14) and hence Theorem 1.1.

1.5. Outline of the paper

In Section 2, we give the proof of Theorem 1.1 by assuming the quantitative version of the Duke’s theorem that is equation (2.1). In Lemma 2.1, we compute the Hardy-Littlewood measure at the archimedean place. In Lemma 2.3, we give an explicit formula for in terms of the quadratic character and then an explicit formula for involving in Lemma 2.4. In Lemma 2.6, we compute the sieve densities defined in equation (1.24). In Lemma 2.7, we give a sharp upper bound on the main term of the Selberg sieve. Finally, we average over these bounds and by proving the average size of these singular series is bounded (analogue of the Gallagher’s theorem) we prove Theorem 1.1.

In Section 3, we prove Theorem 3.1 which implies equation (2.1). Let , and be the the integral points of the orthogonal group of . Then is a lattice and has a natural hyperbolic structure with finitely possible elliptic and cusp posits. We construct an automorphic function defined on from the smooth function . We spectrally expand in the basis of the eigenfunctions of the Laplace-Beltrami operator on We denote the contribution of the constant function by the main term and the contribution of the non-trivial eigenfunctions (Maass cusp forms and Eisenstein series of ) by Er. By assuming Theorem 4.4, that we prove in Section 4, the main term is the product of the local densities. Our goal in Section 3 is to give a power saving upper bound on Er. This power saving in the error term is crucial for the application of the Selberg sieve in Section 2. Let be a positive real number. We write Er as the sum of the low and the hight frequency eigenfunctions in the spectrum

where

(1.27)

and

(1.28)

where is the Weyl sum associated to the eigenfunction ; see equation (3.3). In Section 3.1, we give an upper bound on the contribution of The upper bound follows from the integration by parts and showing that , the Fourier coefficients of the smooth function , decays faster than any polynomial in the spectral parameter . This implies that if for some fixed then . Hence, it suffices to bound the contribution of . In Section 3.2, we prove an explicit form of the Maass identity that relates the Weyl sums to the Fourier coefficients of the associated half weight Maass form obtained by the theta transfer using the Siegel theta Kernel. In Section 3.3, we apply Duke’s non-trivial upper bound on the Fourier coefficients of the weight Maass form and the upper bound on the norm of the theta transfer of a Maass form that we prove in Section 5 to give an upper bound on . There is a technical issue in using Duke’s result. The bound is exponentially growing in the eigenvalues aspect with the term for the weight half eigenfunctions with norm 1 and eigenvalue . We show that this term cancels with the exponentially decaying factor that appears in , the norm of the theta transfer of . This is the content of Section 5. Our method is based on Katok-Sarnak’s approach [KS93]. Biro [Bir00] generalized the work of Sarnak and Katok in the level aspect for with a different method. We generalize the work of Sarnak and Katok in the level aspect for . Therefore, we prove a quantitative version of the euqidistribution of binary quadratic forms of fixed discriminant in Theorem 3.1.

In Section 4, we prove a generalized class number formula in Theorem 4.4. This theorem gives the main term of defined in the equation (2.1). We briefly describe the proof of Theorem 4.4. The proof uses the Siegel Mass formula that gives a product formula for the sum of the representation number of an integer by a quadratic form averaged over the genus class of . In the Lemma 4.1, we show that the genus class of contains only one element for every In the Lemma 4.3, we show that the representation number of each integral point on are equal of where is squarefree. Finally, Theorem 4.4 shows that in fact the Siegel Mass formula gives a product formula for the number of the integral orbits of the orthogonal group on the quadric

In Section 5, we give an upper bound on the norm of the theta transfer of a weight Maass form in the eigenvalue and the level aspect up to a polynomial in these parameters. In Lemma 5.1, we compute the Mellin-transform of the theta lift of by a see-saw identity that is originally due Niwa[Niw75] and used by Sarnak and Katok [KS93]. The see-saw idenity in this case identifies the Mellin transform with the inner product of an Eisenstein series against the product of the weight 1/2 modular form and the complex conjugate of the Jacobi theta series . The last integral against Eisenstein series is explicitly computable by unfolding the Eisenstein series. Hence, we obtain the Fourier coefficients of the theta transfer at the cusp at infinity. Finally, we bound the norm of a modular form by bounding the truncated sum of the squares of its Fourier coefficients; see [Iwa02a, Page 110, equation 8.17]. Note that the norm of the theta transfer of a new form is given by the Rallis-Inner product formula. Since we also deal with old forms, we rather use a more direct approach. We used the classical Seigel theta kernel in order to lift Maass forms into weight 1/2 modular forms and vise versa.

1.6. Acknowledgements

I would like to thank professor Heath-Brown, Radziwill and Soundararajan my mentors at MSRI for several insightful and inspiring conversations during the Spring 2017 at MSRI. In fact, Theorem 1.1 is inspired by the ideas that were developed in my discussions with professor Heath-Brown and professor Radziwill and Soundararajan kindly outlined the proof of Lemma 2.7. Furthermore, I would like to thank Professor Rainer Schulze-Pillot for his comments regarding Siegel Mass formula. I am also very thankful to professor Peter Sarnak, Simon Marshall, and Asif Ali Zaman for their comments and encouragement. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.

2. Generalized Minkowski’s bound for prime ideals of

In this section, we give the proof of Theorem 1.1 by assuming

(2.1)

with a power saving bound on Er. We prove this identity in Theorem 3.1 which is the quantitative version of the Duke’s theorem. We proceed by computing the local densities and

2.1. Selberg upper bound Sieve

We begin by computing explicitly.

Lemma 2.1.

Let be as above in equation (1.23). We have

(2.2)

where

(2.3)
Proof.

We change the variables to and then

Next, we scale the coordinates by and define , and . Hence,

where and

Let

(2.4)

Then

(2.5)

It follows that is a smooth and bounded function where the is bounded by a constant that only depends on the smooth function

Next, we compute explicitly, the local density at each odd prime . We have

where and

where runs over integers modulo with , and runs over vectors in modulo . Since is an odd prime number, we can diagonalize our quadratic form over the local ring by changing the variables to , and and obtain

We apply the following lemma for the computation of local densities. For another versions for this lemma see; [TS17, Lemma 3.1] and Blomer [Blo08, (1.8)].

Lemma 2.2.

Let

where with and with . Assume that where with . Let be the following quadric

defined over Then

(2.6)

where

Moreover if is odd, then

(2.7)

where denote the Legendre symbol of modulo , and if is even then

(2.8)
Proof.

We compute

where , , , , and . We note that the last summation is a Gauss sum. Let be the Gauss sum, and let if and if . Then if , we have

where is the Jacobi symbol. We define when . We have

We substitute the values of and obtain