Some Density Results on Sets of Primes for Hecke Eigenvalues

Let f and g be two distinct holomorphic cusp forms for SL 2 ( Z ) , and we write λ f ( n ) and λ g ( n ) for their corresponding Hecke eigenvalues. Firstly, we study the behavior of the signs of the sequences λ f ( p ) λ f ( p j ) 􏽮 􏽯 for any even positive integer j . Moreover, we obtain the analytic density for the set of primes where the product λ f ( p i ) λ f ( p j ) is strictly less than λ g ( p i ) λ g ( p j ) . Finally, we investigate the distribution of linear combinations of λ f ( p j ) and λ g ( p j ) in a given interval. These results generalize


Introduction
Let H * k be the set of all normalized Hecke primitive cusp forms of even integral weight k ≥ 2 for the full modular group SL 2 (Z) denoted by λ f (n) the n-th Hecke eigenvalue of f ∈ H * k . e Hecke eigenvalues of cusp forms have been extensively studied (see, e.g., [1][2][3][4][5][6][7]). From the theory of Hecke operators, we know that λ f (n) satisfies the standard Hecke relation as follows: for any integers m, n ≥ 1, In particular, λ f (n) is a real and multiplicative function and Furthermore, it is also known that λ f (n) satisfies the Ramanujan conjecture [8]: where d(n) is the Dirichlet divisor function. ere are many papers that focus on the sign changes of the Hecke eigenvalues λ f (n). It is well known that λ f (n) n ≥ 1 changes sign infinitely often. Meher et al. [9] studied the distribution of the signs of λ f (p j ) as p varies over the prime numbers. ey calculated the natural density of the sets explicitly in terms of the celebrated Sato-Tate conjecture (see eorem 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 p|λ f (p j )λ g (p j ) < 0 has natural density 1/2 for any odd positive integer j.
In this paper, based on the now-proven Sato-Tate conjecture, we first study the behavior of the signs of λ f (p)λ f (p j ) for any even positive integer j.
Theorem 1. Let f ∈ H * k be a cusp form. en, for any even positive integer j, the sets p|λ f (p)λ f (p j ) > 0 and p|λ f (p)λ f (p j ) < 0 both have natural density 1/2.
In [13], Kowalski et al. first proved that if the signs of λ f (p) and λ g (p) coincide for all primes up to the exceptional set of analytic density at most 1/32, then f � g. 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 λ f (p) < λ g (p) and λ 2 f (p) < λ 2 g (p) both have analytic density at least 1/16. Notice that the pair-Sato-Tate conjecture yields a stronger result for the former set in [15] with natural density 1/2 in replace of at least 1/16 (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.
It is natural to ask whether eorem 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 λ f (p j ) and λ g (p j ) in a specified interval. Chiriac and Jorza (see Proposition 5.9 in [18]) obtained the density bound for the set v|a < c 1 a v (π 1 ) + c 2 a v (π 2 ) < b in the context of unitary cuspidal representations π that satisfy the Ramanujan conjecture. For holomorphic cusp forms, we establish the following theorem. where e proofs of eorems 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 ( eorem A in [19]), which implies that the L-function L(sym j f, s) is automorphic for j ≥ 1 and f ∈ H * k . en, with the help of the properties of symmetric power L-functions and their Rankin-Selberg L-functions, we obtain the desired results.
Let f, g ∈ H * k be two cusp forms. e j-th symmetric power L-function attached to f is defined by where α f (p) and β f (p) are two complex numbers with 2 Journal of Mathematics One can write it as a Dirichlet series: for Re(s) > 1, where λ sym j f (n) is a real multiplicative function, and e Rankin-Selberg L-function attached to sym i f and sym j g is defined by where λ sym i f×sym j g (n) is a real multiplicative function, and We make the convention that A key ingredient of proving eorems 2 and 3 is the analytic properties of various automorphic L-functions. By a series of deep works [29][30][31][32][33][34][35][36], we learn that for 1 ≤ j ≤ 8, L(sym j f, s) is an automorphic L-function. Recently, Newton and orne (see eorem A in [19]) proved the automorphy of the symmetric power lifting sym j f for j ≥ 1 and f ∈ H * k . Hence, by standard arguments, we have the following.
Lemma 1. Let f ∈ H * k be a cusp form and L(sym j f, s) be defined as in (8). For j ≥ 1, L(sym j f, s) has an analytic continuation as an entire function in the whole complex plane C.
Lemma 2. Let f, g ∈ H * k be two cusp forms and L(sym i f × sym j g, s) be defined as in (12). For i, j ≥ 1, L(sym i f × sym j g, s) has an analytic continuation as an entire function in the whole complex plane C (except possibly for simple poles at s � 0, 1 when sym j f � sym j g).
us, when f � g, s � 1 is a simple pole of L(sym j f × sym j g, s). By (12), we have In other cases, L(sym i f × sym j g, s) is an entire function and does not vanish at s � 1. us, 2.2. Sato-Tate Conjecture. Firstly, let us introduce the definition of natural density.
Definition 1. For a subset A⊆P which denotes the set of all primes, the natural density of A in P is defined as provided the limit exists.
Secondly, let us define the Sato-Tate measure and state the Sato-Tate conjecture (see eorem 2.3 in [9]), which will be used to prove eorem 1.
For any subinterval I⊆[0, π], one has Lemma 3 implies that if A is a finite set, then the natural density of the set p|θ p ∈ A is 0.

An Analytic Density Lemma.
We also recall the definition of analytic density.

Definition 3. A set B of primes is said to have analytic density (or Dirichlet density) δ > 0 if and only if,
In order to prove eorems 2 and 3, we need the following lemma, which is inspired by the ideas outlined in Section 3 of [5].

Proof of Theorem 1
By (9), for any prime p, we can write, for some θ p ∈ [0, π]. And, λ f (p j ) is expressible by the following elementary trigonometric formula: When the values of θ p are 0 or π, the values of λ f (p j ) are j + 1 or (− 1) j (j + 1). en, we have Since the set p|θ p � 0 or π has natural density 0 which only has finitely many primes, we may assume that θ p ∈ (0, π). For θ p ∈ (0, π), we know that sin θ p > 0. Hence, the sign of λ f (p)λ f (p j ) is the same as the sign of cos θ p sin((j + 1)θ p ). e proof of eorem 1 can be divided into two cases when j ≡ 2(mod 4) and j ≡ 0(mod 4).
Next, we consider the sets A and B consisting of the following forms: We will prove the results in eorem 1 by showing that the Sato-Tate measure of the two sets A and B is equal, i.e., we obtain Journal of Mathematics 5 On the other hand, we have J′ sin 2 x dx � π 4(j + 1) We know that . (38) us, the second term on the right-hand side of equations (36) and (37) is same. erefore, we see that Let us prove μ ST (K m ) � μ ST (K m ′ ). 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. j ≡ 0(mod 4). Assume j ≡ 0(mod 4). We see that Obviously, 6 Journal of Mathematics Next, we define the right-hand side of (44) and (45) as e rest of the proof runs as before.

Proof of Theorem 2
e next two lemmas are generalizations of Lemmas 1 and 2 and Lemmas 3.1 and 3.2 of Lao [17], respectively.
Lemma 5. Let f, g ∈ H * k be two distinct cusp forms. en, for j ≥ 1 and 0 ≤ i ≤ j, we have where δ 1 is defined as in (4).
□ Data Availability e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.