Some Density Results on Sets of Primes for Hecke Eigenvalues

Zou, Aiyue; Lao, Huixue; Luo, Shu

doi:https://doi.org/10.1155/2021/2462693

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Volume 2021 | Article ID 2462693 | https://doi.org/10.1155/2021/2462693

Some Density Results on Sets of Primes for Hecke Eigenvalues

Aiyue Zou,¹Huixue Lao,¹and Shu Luo¹

Academic Editor: Jie Wu

Received03 Jun 2021

Accepted26 Jun 2021

Published22 Jul 2021

Abstract

Let and be two distinct holomorphic cusp forms for , and we write and for their corresponding Hecke eigenvalues. Firstly, we study the behavior of the signs of the sequences for any even positive integer . Moreover, we obtain the analytic density for the set of primes where the product is strictly less than . Finally, we investigate the distribution of linear combinations of and in a given interval. These results generalize previous ones.

1. Introduction

Let be the set of all normalized Hecke primitive cusp forms of even integral weight for the full modular group denoted by the n-th Hecke eigenvalue of . The Hecke eigenvalues of cusp forms have been extensively studied (see, e.g., [1–7]). From the theory of Hecke operators, we know that satisfies the standard Hecke relation as follows: for any integers ,

In particular, is a real and multiplicative function and

Furthermore, it is also known that satisfies the Ramanujan conjecture [8]:where is the Dirichlet divisor function.

There are many papers that focus on the sign changes of the Hecke eigenvalues . It is well known that changes sign infinitely often. Meher et al. [9] studied the distribution of the signs of as varies over the prime numbers. They calculated the natural density of the sets explicitly in terms of the celebrated Sato–Tate conjecture (see Theorem B in [10]). In [11], a joint version of the pair-Sato–Tate conjecture (as outlined in Proposition 2.2 in [12]) gives the result that the set has natural density for any odd positive integer .

In this paper, based on the now-proven Sato–Tate conjecture, we first study the behavior of the signs of for any even positive integer .

Theorem 1. Let be a cusp form. Then, for any even positive integer , the sets and both have natural density .

In [13], Kowalski et al. first proved that if the signs of and coincide for all primes up to the exceptional set of analytic density at most , then . Subsequently, Matomäki [14] improved the above result by utilizing linear programming to take full advantage of all the available information.

Inspired by [13], Chiriac [15] started to compare Hecke eigenvalues over prime numbers and simultaneously showed that the sets of primes for and both have analytic density at least . Notice that the pair-Sato–Tate conjecture yields a stronger result for the former set in [15] with natural density in replace of at least (see Proposition 2.1 (iii) in [16]). Of course, this result is also valid for the analytic density since the existence of the natural density implies that of the analytic density, and they are equal.

Most recently, Lao [17] further studied Chiriac’s questions [15] by considering the Hecke eigenvalues at prime powers. She established that the sets with and with have analytic density at least and , respectively.

In this paper, our second aim is to obtain the analytic density for the set with and .

Theorem 2. Let be two distinct cusp forms. Then, for and , the set has analytic density at least , where

Remark 1. By symmetry, for and , Theorem 2 implies that the set has analytic density at least . Thus, for and , the set has analytic density at most .

Remark 2. One can easily find that for , or , we have , which are consistent with the numerical values of Chiriac’s results (see Theorems 1 and 1.3 in [15]). By comparison, with , and with . Hence, Theorem 2 also generalizes previous results of Lao (see Theorems 1 and 2 in [17]).

It is natural to ask whether Theorem 2 can be refined by the pair-Sato–Tate conjecture. In fact, with the help of this conjecture, some specific cases can be dealt with by calculating the corresponding double integral.

Finally, we concern the distribution of linear combinations of and in a specified interval. Chiriac and Jorza (see Proposition 5.9 in [18]) obtained the density bound for the set in the context of unitary cuspidal representations that satisfy the Ramanujan conjecture. For holomorphic cusp forms, we establish the following theorem.

Theorem 3. Let be two distinct cusp forms. Let with . Then, for , the set has analytic density at leastwhere

The proofs of Theorems 2 and 3 rely on Lemma 4 involving the analytic density of a particular set of primes. Recently, there was a big breakthrough on the automorphy of all symmetric powers for cuspidal Hecke eigenforms (Theorem A in [19]), which implies that the -function is automorphic for and . Then, with the help of the properties of symmetric power -functions and their Rankin–Selberg -functions, we obtain the desired results.

2. Preliminaries

2.1. Primitive Automorphic -Functions

In this section, we will briefly recall some fundamental facts about primitive automorphic -functions and give the main tools and definitions. For more details to learn automorphic -functions, refer [17, 20–28].

Let be two cusp forms. The -th symmetric power -function attached to is defined bywhere and are two complex numbers with

One can write it as a Dirichlet series: for ,where is a real multiplicative function, and

The Rankin–Selberg -function attached to and is defined bywhere is a real multiplicative function, and

We make the convention that

A key ingredient of proving Theorems 2 and 3 is the analytic properties of various automorphic -functions. By a series of deep works [29–36], we learn that for , is an automorphic -function. Recently, Newton and Thorne (see Theorem A in [19]) proved the automorphy of the symmetric power lifting for and . Hence, by standard arguments, we have the following.

Lemma 1. Let be a cusp form and be defined as in (8). For , has an analytic continuation as an entire function in the whole complex plane .

Combining Lemma 1 with (10), we deduce that for ,

Moreover, based on the automorphy of for and the work [37–42] on the Rankin–Selberg theory, we have the following.

Lemma 2. Let be two cusp forms and be defined as in (12). For , has an analytic continuation as an entire function in the whole complex plane (except possibly for simple poles at when ).

Thus, when , is a simple pole of . By (12), we have

In other cases, is an entire function and does not vanish at . Thus,

2.2. Sato–Tate Conjecture

Firstly, let us introduce the definition of natural density.

Definition 1. For a subset which denotes the set of all primes, the natural density of in is defined asprovided the limit exists.

Secondly, let us define the Sato–Tate measure and state the Sato–Tate conjecture (see Theorem 2.3 in [9]), which will be used to prove Theorem 1.

Definition 2. The Sato–Tate measure is the probability measure on given by .

For any subinterval , one has

Lemma 3. Let be a cusp form. The sequence is equidistributed in with respect to the Sato–Tate measure . In particular, for any subinterval , we have

Lemma 3 implies that if is a finite set, then the natural density of the set is 0.

2.3. An Analytic Density Lemma

We also recall the definition of analytic density.

Definition 3. A set of primes is said to have analytic density (or Dirichlet density) if and only if,

In order to prove Theorems 2 and 3, we need the following lemma, which is inspired by the ideas outlined in Section 3 of [5].

Lemma 4. Let be real numbers only determined by prime and satisfy with a bound B that does not depend on . There exist real constants and such that as , the following equalities hold:

Then, the set has analytic density at least .

Proof. If , , and therefore , whereas if , then and . We see thatInserting (23) into (24), we getComparing (22) with (25) leads toi.e., the set has analytic density at least .

3. Proof of Theorem 1

By (9), for any prime , we can write,for some . And, is expressible by the following elementary trigonometric formula:

When the values of are 0 or , the values of are or .

Then, we have

Since the set has natural density 0 which only has finitely many primes, we may assume that . For , we know that . Hence, the sign of is the same as the sign of .

The proof of Theorem 1 can be divided into two cases when and .

3.1.

Assume that , we get

Then, we observe that

Especially, we rewrite the third term of the right-hand side of (31) and (32) by using (30).

Next, we consider the sets and consisting of the following forms:

We will prove the results in Theorem 1 by showing that the Sato–Tate measure of the two sets and is equal, i.e., . Since the sets and are divided into disjoint intervals given in (34), it is sufficient to prove that for each with and for each with and .

Now, we restrict our attention to prove . Using the factwe obtain

On the other hand, we have

We know that

Thus, the second term on the right-hand side of equations (36) and (37) is same. Therefore, we see that

Let us prove . Applying (35) again yields the following equations:

It is easy to find that

From the above discussion, we deduce that

Similarly, we can have

3.2.

Assume . We see that

Obviously,

Next, we define the right-hand side of (44) and (45) as

The rest of the proof runs as before.

4. Proof of Theorem 2

The next two lemmas are generalizations of Lemmas 1 and 2 and Lemmas 3.1 and 3.2 of Lao [17], respectively.

Lemma 5. Let be two distinct cusp forms. Then, for and , we havewhere is defined as in (4).

Proof. By observing (11) and (13), we find that for ,In particular, for ,Hence, we haveBy (51) and (52), we see that for and ,Note that using (17), for and , we havewhere is defined as in (4). In particular, taking , we apply (2) and getNow, using (15)–(17) and (54) to estimate the terms in (53), we obtain Lemma 5 immediately.

Lemma 6. Let be two distinct cusp forms. Then, for and , we havewhere is defined as in (5).

Proof. By (49), we know that for ,

Case 1. If is odd, then so is . Then, (11) givesSince (see (9)), we have

Case 2. If is even, then so is . Similarly, we haveThus,By the above estimates, we conclude that for and ,where is defined as in (5).

Now, we are ready to prove Theorem 2.

Proof. of Theorem 2. Let be two distinct cusp forms. For and , defineWe infer from (15) and (49) thatObserve that .
Moreover, we know from Lemmas 5 and 6 that and .
Therefore, from the proof of Lemma 4, we havei.e., the set has analytic density at least .

5. Proof of Theorem 3

The following lemmas play an important role in the Proof of Theorem 3.

Lemma 7. Let be two distinct cusp forms. Let with . Then, for and , we havewhere is given in (7).

Proof. It is easy to verify that the sum on the left side of (66) equalsFor the first sum in (67), by using (15), we getFor the second sum in (67), one can easily obtain from (15), (17), and (50) thatAs for the third sum in (67), we note thatIn view of (50), we haveNoticing (11), (13), and (49), we clearly deduce thatIf is even and , we apply (2) and getHence, we deduce from (15) and (17) that as ,where is given in (7).
We now calculate the fourth sum in (67). By (50), we haveCombining the above result with (15) and (16) leads toFrom (15), (17), and (72), it is easy to check that as ,Moreover, we infer from (15), (17), and (52) thatBy the above results, it is shown that as ,We finish the proof by inserting (68), (69), (74), and (79) into (67).

Lemma 8. Let be two distinct cusp forms. Let with . Then, for , we have

Proof. It follows from (9) and (11) thatBy similar computations, we infer from (13) and (50) thatFor , we haveAccording to the triangle inequality, we draw the conclusion.

Proof. of Theorem 3. Let be two distinct cusp forms. It is easy to verify from (15), (17), (50), and (83) thatCombining the above result with Lemmas 7 and 8, Theorem 3 follows apparently by applying Lemma 4 with

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Natural Science Foundation of Shandong Province (Grant no. ZR2018MA003).

References

G. Lü, “The sixth and eighth moments of fourier coefficients of cusp forms,” Journal of Number Theory, vol. 129, no. 11, pp. 2790–2800, 2009.
View at: Publisher Site | Google Scholar
G. Lü, “On general divisor problems involving hecke eigenvalues,” Acta Mathematica Hungarica, vol. 135, no. 1-2, pp. 148–159, 2012.
View at: Publisher Site | Google Scholar
G. Lü, “Shifted convolution sums of fourier coefficients with divisor functions,” Acta Mathematica Hungarica, vol. 146, no. 1, pp. 86–97, 2015.
View at: Publisher Site | Google Scholar
H. F. Liu and R. Zhang, “Some problems involving hecke eigenvalues,” Acta Mathematica Hungarica, vol. 159, no. 1, pp. 287–298, 2019.
View at: Publisher Site | Google Scholar
Y.-K. Lau and G. Lü, “Sums of fourier coefficients of cusp forms,” The Quarterly Journal of Mathematics, vol. 62, no. 3, pp. 687–716, 2011.
View at: Publisher Site | Google Scholar
J. Wu and W. Zhai, “Distribution of hecke eigenvalues of newforms in short intervals,” The Quarterly Journal of Mathematics, vol. 64, no. 2, pp. 619–644, 2013.
View at: Publisher Site | Google Scholar
D. Zhang, Y.-K. Lau, and Y. Wang, “Remark on the paper “on products of fourier coefficients of cusp forms”,” Archiv der Mathematik, vol. 108, no. 3, pp. 263–269, 2017.
View at: Publisher Site | Google Scholar
P. Deligne, “La conjecture de Weil. I,” Publications Mathématiques de l’IHÉS, vol. 43, no. 1, pp. 273–307, 1974.
View at: Publisher Site | Google Scholar
J. Meher, K. D. Shankhadhar, and G. K. Viswanadham, “On the coefficients of symmetric power L-functions,” International Journal of Number Theory, vol. 14, no. 3, pp. 813–824, 2018.
View at: Publisher Site | Google Scholar
T. Barnet-Lamb, D. Geraghty, M. Harris, and R. Taylor, “A family of calabi-yau varieties and potential automorphy II,” Publications of the Research Institute for Mathematical Sciences, vol. 47, pp. 29–98, 2011.
View at: Publisher Site | Google Scholar
M. A. Amri, “Simultaneous sign change and equidistribution of signs of fourier coefficients of two cusp forms,” Archiv der Mathematik, vol. 111, no. 3, pp. 257–266, 2018.
View at: Publisher Site | Google Scholar
P.-J. Wong, “On the chebotarev-sato-tate phenomenon,” Journal of Number Theory, vol. 196, pp. 272–290, 2019.
View at: Publisher Site | Google Scholar
E. Kowalski, Y.-K. Lau, K. Soundararajan, and J. Wu, “On modular signs,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 149, no. 3, pp. 389–411, 2010.
View at: Publisher Site | Google Scholar
K. Matomäki, “On signs of fourier coefficients of cusp forms,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 152, no. 2, pp. 207–222, 2012.
View at: Publisher Site | Google Scholar
L. Chiriac, “Comparing hecke eigenvalues of newforms,” Archiv der Mathematik, vol. 109, no. 3, pp. 223–229, 2017.
View at: Publisher Site | Google Scholar
L. Chiriac, “On the number of dominating fourier coefficients of two newforms,” Proceedings of the American Mathematical Society, vol. 146, no. 10, pp. 4221–4224, 2018.
View at: Publisher Site | Google Scholar
H. X. Lao, “On comparing hecke eigenvalues of cusp forms,” Acta Mathematica Hungarica, vol. 160, no. 1, pp. 58–71, 2020.
View at: Publisher Site | Google Scholar
L. Chiriac and A. Jorza, “Comparing hecke coefficients of automorphic representations,” Transactions of the American Mathematical Society, vol. 372, no. 12, pp. 8871–8896, 2019.
View at: Publisher Site | Google Scholar
J. Newton and J. A. Thorne, “Symmetric power functoriality for holomorphic modular forms,” 2020, http://arxiv.org/abs/1912.11261v2.
View at: Google Scholar
L. Dong, H. Liu, and D. Zhang, “Zero density estimates for automorphic L-functions of GL_m,” Acta Mathematica Hungarica, vol. 148, no. 1, pp. 191–210, 2016.
View at: Publisher Site | Google Scholar
H. Iwaniec, Topics in Classical Automorphic Forms, American Mathematical Society, Providence, RI, USA, 1997.
Y. Jiang, G. Lü, and X. Yan, “Mean value theorem connected with fourier coefficients of hecke-maass forms for SL (m, ℤ),” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 161, no. 2, pp. 339–356, 2016.
View at: Publisher Site | Google Scholar
H.-F. Liu, “Mean value estimates of the coefficients of product L-functions,” Acta Mathematica Hungarica, vol. 156, no. 1, pp. 102–111, 2018.
View at: Publisher Site | Google Scholar
H. Lao, M. McKee, and Y. Ye, “Asymptotics for cuspidal representations by functoriality from GL (2),” Journal of Number Theory, vol. 164, pp. 323–342, 2016.
View at: Publisher Site | Google Scholar
H. Lao and A. Sankaranarayanan, “The distribution of fourier coefficients of cusp forms over sparse sequences,” Acta Arithmetica, vol. 163, no. 2, pp. 101–110, 2014.
View at: Publisher Site | Google Scholar
H. Lao and H. Wei, “Ω-result on coefficients of automorphic L-functions over sparse sequences,” Journal of the Korean Mathematical Society, vol. 52, no. 5, pp. 945–954, 2015.
View at: Publisher Site | Google Scholar
P. Song, W. Zhai, and D. Zhang, “Power moments of hecke eigenvalues for congruence group,” Journal of Number Theory, vol. 198, pp. 139–158, 2019.
View at: Publisher Site | Google Scholar
D. Zhang and Y. Wang, “Higher-power moments of Fourier coefficients of holomorphic cusp forms for the congruence subgroup Γ₀ (N),” The Ramanujan Journal, vol. 47, no. 3, pp. 685–700, 2018.
View at: Publisher Site | Google Scholar
L. Clozel and J. A. Thorne, “Level-raising and symmetric power functoriality, I,” Compositio Mathematica, vol. 150, no. 5, pp. 729–748, 2014.
View at: Publisher Site | Google Scholar
L. Clozel and J. A. Thorne, “Level-raising and symmetric power functoriality. II,” Annals of Mathematics, vol. 181, no. 2, pp. 303–359, 2015.
View at: Publisher Site | Google Scholar
L. Clozel and J. A. Thorne, “Level-raising and symmetric power functoriality, III,” Duke Mathematical Journal, vol. 166, pp. 325–402, 2017.
View at: Publisher Site | Google Scholar
L. Dieulefait, “Automorphy of Symm⁵ (GL(2)) and base change,” Journal de Mathématiques Pures et Appliquées, vol. 104, no. 4, pp. 619–656, 2015.
View at: Publisher Site | Google Scholar
S. Gelbart and H. Jacquet, “A relation between automorphic representations of GL (2) and GL (3),” Annales Scientifiques de l’École Normale Supérieure, vol. 11, no. 4, pp. 471–542, 1978.
View at: Publisher Site | Google Scholar
H. H. Kim, “Functoriality for the exterior square of and the symmetric fourth of . appendix 1 by D. Ramakrishnan and appendix 2 by H. H. Kim and P. Sarnak,” Journal of the American Mathematical Society, vol. 16, pp. 139–183, 2003.
View at: Publisher Site | Google Scholar
H. H. Kim and F. Shahidi, “Functorial products for GL₂ × GL₃ and the symmetric cube for GL₂,” The Annals of Mathematics, vol. 155, no. 3, pp. 837–893, 2002.
View at: Publisher Site | Google Scholar
H. H. Kim and F. Shahidi, “Cuspidality of symmetric powers with applications,” Duke Mathematical Journal, vol. 112, pp. 177–197, 2002.
View at: Publisher Site | Google Scholar
H. Jacquet and J. A. Shalika, “On euler products and the classification of automorphic representations I,” American Journal of Mathematics, vol. 103, no. 3, pp. 499–558, 1981.
View at: Publisher Site | Google Scholar
H. Jacquet and J. A. Shalika, “On euler products and the classification of automorphic forms II,” American Journal of Mathematics, vol. 103, no. 4, pp. 777–815, 1981.
View at: Publisher Site | Google Scholar
Y.-K. Lau and J. Wu, “A density theorem on automorphic L-functions and some applications,” Transactions of the American Mathematical Society, vol. 358, pp. 441–472, 2006.
View at: Publisher Site | Google Scholar
Z. Rudnick and P. Sarnak, “Zeros of principal -functions and random matrix theory,” Duke Mathematical Journal, vol. 81, pp. 269–232, 1996.
View at: Publisher Site | Google Scholar
F. Shahidi, “On certain L-functions,” American Journal of Mathematics, vol. 103, no. 2, pp. 297–355, 1981.
View at: Publisher Site | Google Scholar
F. Shahidi, “Third symmetric power -functions for ,” Compositio Mathematica, vol. 70, pp. 245–273, 1989.
View at: Google Scholar

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Copyright © 2021 Aiyue Zou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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