Denote by aK(n) the number of integral ideals in K with norm n, where K is a algebraic number field of degree m over the rational field Q. Let p be a prime number. In this paper, we prove that, for two distinct quadratic number fields Ki=Q(√di), i=1,2, the sets both
{p | aK1(p)<aK2(p)} and {p | aK1(p2)<aK2(p2)}
have analytic density 1/4, respectively.
Citation: Qian Wang, Xue Han. Comparing the number of ideals in quadratic number fields[J]. Mathematical Modelling and Control, 2022, 2(4): 268-271. doi: 10.3934/mmc.2022025
[1] | Moumita Bhattacharyya, Shib Sankar Sana . A second order quadratic integral inequality associated with regular problems. Mathematical Modelling and Control, 2024, 4(1): 141-151. doi: 10.3934/mmc.2024013 |
[2] | Yameng Zhang, Xia Zhang . On the fractional total domatic numbers of incidence graphs. Mathematical Modelling and Control, 2023, 3(1): 73-79. doi: 10.3934/mmc.2023007 |
[3] | Guoyi Li, Jun Wang, Kaibo Shi, Yiqian Tang . Some novel results for DNNs via relaxed Lyapunov functionals. Mathematical Modelling and Control, 2024, 4(1): 110-118. doi: 10.3934/mmc.2024010 |
[4] | Archit Chaturvedi . Polymer physics-based mathematical models for the correlation of DNA and mRNA in a eukaryotic cell. Mathematical Modelling and Control, 2022, 2(3): 131-138. doi: 10.3934/mmc.2022014 |
[5] | Bharatkumar K. Manvi, Shravankumar B. Kerur, Jagadish V Tawade, Juan J. Nieto, Sagar Ningonda Sankeshwari, Hijaz Ahmad, Vediyappan Govindan . MHD Casson nanofluid boundary layer flow in presence of radiation and non-uniform heat source/sink. Mathematical Modelling and Control, 2023, 3(3): 152-167. doi: 10.3934/mmc.2023014 |
[6] | Yunsi Yang, Jun-e Feng, Lei Jia . Recent advances of finite-field networks. Mathematical Modelling and Control, 2023, 3(3): 244-255. doi: 10.3934/mmc.2023021 |
[7] | Iman Malmir . Novel closed-loop controllers for fractional nonlinear quadratic systems. Mathematical Modelling and Control, 2023, 3(4): 345-354. doi: 10.3934/mmc.2023028 |
[8] | Daizhan Cheng, Zhengping Ji, Jun-e Feng, Shihua Fu, Jianli Zhao . Perfect hypercomplex algebras: Semi-tensor product approach. Mathematical Modelling and Control, 2021, 1(4): 177-187. doi: 10.3934/mmc.2021017 |
[9] | Eminugroho Ratna Sari, Nikken Prima Puspita, R. N. Farah . Prevention of dengue virus transmission: insights from host-vector mathematical model. Mathematical Modelling and Control, 2025, 5(2): 131-146. doi: 10.3934/mmc.2025010 |
[10] | Erick M. D. Moya, Diego Samuel Rodrigues . A mathematical model for the study of latent tuberculosis under 3HP and 1HP regimens. Mathematical Modelling and Control, 2024, 4(4): 400-416. doi: 10.3934/mmc.2024032 |
Denote by aK(n) the number of integral ideals in K with norm n, where K is a algebraic number field of degree m over the rational field Q. Let p be a prime number. In this paper, we prove that, for two distinct quadratic number fields Ki=Q(√di), i=1,2, the sets both
{p | aK1(p)<aK2(p)} and {p | aK1(p2)<aK2(p2)}
have analytic density 1/4, respectively.
Suppose that K is an algebraic number field of degree m over the rational field Q. The Dedekind zeta function ζK(s) is defined as, for Re s>1,
ζK(s)=∑a1N(a)s, |
where a varies over the integral ideals of K and N(a) denotes the norm of a. Denote by aK(n) the number of integral ideals in K with norm n, then we can rewrite ζK(s) as
ζK(s)=∞∑n=1aK(n)ns,Re s>1. |
The arithmetic function aK(n) is one of the research hotspots in algebraic number theory, since its behavior is not regular. It is known from Chandraseknaran and Good [1] that aK(n) is multiplicative and satisfies the upper bound
aK(n)≤d(n)m, | (1.1) |
where d(n) is the divisor function and m=[K:Q].
Let Sk(N)new be the space of all cuspidal newforms of even integral weight k for the congruence subgroup Γ0(N)⊆SL2(Z), with trivial nebentypus. Let f∈Sk(N)new be a Hecke eigenform, and denote by λf(n) the corresponding normalized Hecke eigenvalues, which are studied by many scholars (see [2,3,4,5,6,7,8] etc.). In particular, Chiriac [9] proposed an interesting problem: Is it possible that for two distinct newforms f and g the eigenvalues λf(p) are not less than λg(p), for almost all primes p? To state Chiriac's result, we say that a set Ξ of primes has analytic density (or Dirichlet density) δ>0 if and only if
∑p∈Ξ1ps∼δ∑p1ps,as s→1+. | (1.2) |
Chiriac [9] showed that the problem he proposed cannot occur by proving that for two distinct cusp forms, the set
{p | λf(p)<λg(p)} |
has analytic density at least 1/16. In the same paper, assuming that f and g do not have complex multiplication, and that neither is a quadratic twist of the other, Chiriac [9] also proved that the set
{p | λ2f(p)<λ2g(p)} |
has analytic density at least 1/16. Later, some more results were established in this direction, and we refer to the references [10,11,12] for details.
Motivated by the above works, naturally we draw our attention to the following question: Is it possible that for two distinct quadratic number fields K1 and K2 the number of integral ideals aK1(p) are not less than aK2(p), for almost all primes p? We are able to show that the answer to above question is negative by proving the following result.
Theorem 1.1. Let Ki=Q(√di), i=1,2 be two quadratic number fields, where d1,d2≠0,1 are two distinct square-free integers. Then the sets both
{p | aK1(p)<aK2(p)} and {p | aK1(p2)<aK2(p2)} |
have analytic density 1/4, respectively.
Since the structure of the quadratic number field is more detailed, we can get a precise density result in Theorem 1.1. One key point of the proof is the famous Čebotarev Density theorem.
The following lemma is the famous Čebotarev Density theorem, which can be found in [13,Theorem 31].
Lemma 2.1. Let K,k be algebraic number fields such that K is Galois over k, let σ be an element of Gal(K/k) and denote by ⟨σ⟩ the conjugacy class of σ. Let S be the set of prime ideals p of k such that for every P above p the Frobenius element [K/kp] lies in ⟨σ⟩. Then S has Dirichlet density card(⟨σ⟩)/card(Gal(K/k)).
For convenience, we write
Ξ1={p | aK1(p)<aK2(p)} and Ξ2={p | aK1(p2)<aK2(p2)}. |
Let K=Q(√d) be a quadratic number field with a square-free integer d≠0,1. It is known that its discriminant is
D={d,if d≡1(mod4),4d,if d≡2,3(mod4). |
Then we have the following proposition.
Proposition 2.2. Let Ki=Q(√di), i=1,2 be two quadratic number fields, where d1,d2≠0,1 are two distinct square-free integers. D1 and D2 are the discriminants of K1 and K2, respectively. Then for p∤D1D2, both p∈Ξ1 and p∈Ξ2 are equivalent to that p is inert in K1 and splits in K2, respectively.
Proof. We first consider the quadratic number field K=Q(√d). Since p∤D, the prime p does not ramify. Thus the prime p either splits or is inert in K.
When p splits in K, then pOK=p1p2 with p1≠p2. Then we have that the ideals with norm p are p1,p2 and the ideals with norm p2 are p21,p1p2,p22. When p is inert in K, the pOK is the only prime ideal with norm p2 of K above p. There are no ideals with norm p.
Thus we have
aK(p)={2,if p splits in K,0,if p is inert in K,aK(p2)={3,if p splits in K,1,if p is inert in K. |
Then from the above two formulas we can get this proposition.
With the help of Čebotarev Density theorem and Proposition 2.2, we can complete the proof of Theorem 1.1.
Note that in the quadratic number field K, p ramifies if and only if p∣D. Thus the number of p which ramifies in K is limited. We just need to focus on the case that p does not ramifies. From Proposition 2.2, it is sufficient to prove that the density of primes which are inert in K1 and split in K2 is 1/4.
Let S=K1K2=Q(√d1,√d2). Due to the fact that d1,d2≠0,1 are two distinct square-free integers, we have
K1∩K2=Q,Gal(S/Q)≅Gal(K1/Q)×Gal(K2/Q). |
We know that S is the splitting field of the polynomial (x2−d1)(x2−d2) with roots ω1=√d1,ω2=−√d1,ω3=√d2,ω4=−√d2. By the ordering of roots we can identify Gal(S/Q) with the permutation group
V4={id,(1,2),(3,4),(1,2)(3,4)}. |
For a prime p, we write Frobp as Frobenius element of Gal(S/Q)≅V4 corresponding to p. Then p is inert in K1 and splits in K2 if and only if Frobp=(1,2). From Lemma 2.1, we know that density of primes which are inert in K1 and split in K2 is 1/4. Therefore, we complete the proof of Theorem 1.1.
Let K be a algebraic number field and suppose that aK(n) denotes the number of integral ideals in K with norm n. In this paper, for the question: Is it possible that for two distinct quadratic number fields K1 and K2 the number of integral ideals aK1(p) are not less than aK2(p), for almost all primes p? We give a negative answer and further show that the sets both
{p | aK1(p)<aK2(p)} and {p | aK1(p2)<aK2(p2)} |
have analytic density 1/4, respectively.
This work is supported by National Natural Science Foundation of China (Grant No. 12171286).
The authors declare there is no conflict of interest in this paper.
[1] |
K. Chandraseknaran, A. Good, On the number of integeral ideals in Galois extensions, Monatshefte für Mathematik, 95 (1983), 99–109. https://doi.org/10.1007/BF01323653 doi: 10.1007/BF01323653
![]() |
[2] |
X. Han, X. Yan, D. Zhang, On fourier coefficients of the symmetric square L-function at Piatetski-Shapiro prime twins, Mathematics, 9 (2021), 1254. https://doi.org/10.3390/math9111254 doi: 10.3390/math9111254
![]() |
[3] |
J. Huang, T. Li, H. Liu, F. Xu, The general two-dimensional divisor problems involving Hecke eigenvalues, AIMS Mathematics, 7 (2022), 6396–6403. https://doi.org/10.3934/math.2022356 doi: 10.3934/math.2022356
![]() |
[4] |
Y. Jiang, G. Lü, X. Yan, Mean value theorem connected with Fourier coefficients of Hecke-Maass forms for SL(m,Z), Mathematical Proceedings of the Cambridge Philosophical Society, 161 (2016), 339–356. https://doi.org/10.1017/S030500411600027X doi: 10.1017/S030500411600027X
![]() |
[5] |
H. Liu, R. Zhang, Some problems involving Hecke eigenvalues, Acta Math. Hung., 159 (2019), 287–298. https://doi.org/10.1007/s10474-019-00913-w doi: 10.1007/s10474-019-00913-w
![]() |
[6] |
P. Song, W. Zhai, D. Zhang, Power moments of Hecke eigenvalues for congruence group, J. Number Theory, 198 (2019), 139–158. https://doi.org/10.1016/j.jnt.2018.10.006 doi: 10.1016/j.jnt.2018.10.006
![]() |
[7] |
D. Zhang, Y. Wang, Ternary quadratic form with prime variables attached to Fourier coefficients of primitive holomorphic cusp form, J. Number Theory, 176 (2017), 211–225. https://doi.org/10.1016/j.jnt.2016.12.018 doi: 10.1016/j.jnt.2016.12.018
![]() |
[8] |
D. Zhang, W. Zhai, On the distribution of Hecke eigenvalues over Piatetski-Shapiro prime twins, Acta Mathematica Sinica, English Series, 37 (2021), 1453–1464. https://doi.org/10.1007/s10114-021-0174-3 doi: 10.1007/s10114-021-0174-3
![]() |
[9] |
L. Chiriac, Comparing Hecke eigenvalues of newforms, Archiv der Mathematik, 109 (2017), 223–229. https://doi.org/10.1007/s00013-017-1072-x doi: 10.1007/s00013-017-1072-x
![]() |
[10] |
L. Chiriac, On the number of dominating Fourier coefficients of two newforms, Proceedings of the American Mathematical Society, 146 (2018), 4221–4224. https://doi.org/10.1090/proc/14145 doi: 10.1090/proc/14145
![]() |
[11] |
L. Chiriac, A. Jorza, Comparing Hecke coefficients of automorphic representations, T. Am. Math. Soc., 372 (2019), 8871–8896. https://doi.org/10.1090/tran/7903 doi: 10.1090/tran/7903
![]() |
[12] |
H. Lao, On comparing Hecke eigenvalues of cusp forms, Acta Math. Hung., 160 (2020), 58–71. https://doi.org/10.1007/s10474-019-00996-5 doi: 10.1007/s10474-019-00996-5
![]() |
[13] | H. P. F. Swinnerton-Dyer, A brief guide to algebraic number theory, London Mathematical Society Student Texts, 50, Cambridge: Cambridge University Press, 2001. https://doi.org/10.1017/CBO9781139173360 |