On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.


Introduction
Let k be an even positive integer, f be a holomorphic cusp form of weight k for the full modular group and λ f (n) be the normalized n-th Fourier coefficient of f , i.e., If we assume that f is an eigenform of all the Hecke operators, then f can be normalized such that λ f (1) = 1 and λ f (n) is real. We define the Hecke L-function associated to f for s > 1 by For any prime p and all integers ν ≥ 0, we have where α f (p), β f (p) are the local parameters of L(s, f ) at prime p, satisfying Then we have λ 2 f (p) = 1 + λ f (p 2 ). Deligne [1,2] proved Ramanujan-Petersson conjecture, i.e., |λ f (n)| ≤ d(n) n ε for all n ≥ 1, where d(n) = ∑ d|n 1.
In order to detect the sign changes of λ f (n), many authors have studied the mean value of λ f (n) and obtained some good results. For example, see [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In addition, the sums of λ f (n) over primes have also been studied. It is known that (see for example Section 5.6 of Iwaniec and Kowalski [23]) there exists a constant C > 0 such that The upper bound of (1) may reach N 1 2 +ε , assuming the Riemann Hypothesis. Furthermore, we can establish that by using the analytic properties of the Rankin-Selberg L-function L(s, f ⊗f ). Another interesting question considered by many authors is the mean value of λ f (n) over certain sets of primes. For example, Baier and Zhao [24] studied the distribution of λ f (n) at Piatetski-Shapiro primes by considering estimates of exponential sum involving Hecke eigenvalue. Moreover, they conjectured that, for 1 < c < 1 + θ with some suitable Furthermore, we can define Piatetski-Shapiro prime twins if p, q are primes and q = [p c ], for any fixed c ∈ R + \ N. Balog [25] and Dufner [26] proved that for almost all c ∈ (0, 1) in the sense of Lebesgue measure. Furthermore, assuming the Riemann Hypothesis of automorphic L-function L(s, f ) is true, they found that (2) holds for all c ∈ (0, 1). Furthermore, Zhang and Zhai [27] studied the mean value of λ f (n) over Piatetski-Shapiro prime twins. Motivated by the above results, we are interested in the distribution of λ 2 f (n) at Piatetski-Shapiro prime twins. For the form f , we know the L(s, sym 2 f ) is an L-function for some GL(3, Z) automorphic representation, which is often called the symmetric-square lift of f . The n-th Fourier coefficient of L(s, sym 2 f ) satisfies The symmetric square L-function associated to f is defined by in the half-plane s > 1. Then, for all n ≥ 1, λ sym 2 f (n) is also multiplicative, real and where d 3 (n) = ∑ m|n d(m). Furthermore, for all primes p, we have Many authors studied the mean value of λ sym 2 f (n). For example, see References [28][29][30][31][32][33][34][35][36]. In this paper, we consider the mean value of Fourier coefficients of symmetric square Lfunction over Piatetski-Shapiro prime twins and obtain the following results, which imply a result on the distribution of λ 2 f (n) at Piatetski-Shapiro prime twins.

Proof. The result follows from Theorem 2 and (2).
Notation. Throughout the paper, ε always denotes a sufficiently small positive constant. Let δ 1 (ε) be sufficiently small and depend on ε. We write f (x) g(x), or f (x) = O(g(x)), to mean that | f (x)| ≤ Cg(x). Let ρ = σ + iη be the nontrivial zero of the symmetric square L-function L(s, sym 2 f ). As usual, Λ(n) is the von Mangoldt function.

Auxiliary Lemmas
Lemma 1. Let η run through a countable set of reals, with c(η) arbitrary complex such that Proof. This lemma is Lemma 1 of Gallagher [37].
This combined with Theorem 1.1, 1.2 of Ye and Zhang [38] and [39] gives this lemma.
Proof. See, for example, Iwaniec and Kowalski [23]. For convenience of calculation, we reduce the summation range of n from n ≤ u to n < u.
Lemma 5. Let L(s, f ) be an L-function of degree k such that Rankin-Selberg convolutions L(s, f ⊗ f ) and L(s, f ⊗f ) exist, and the latter has a simple pole at s = α + it = 1 while the former is entire if f =f . Suppose that the ramified primes |α f (p)| ≤ p/2. There exists an absolute constant c > 0 such that L(s, f ) has no zeros in the region where q( f , s) is the analytic conductor and q( f ) = q( f , 0).
where β = β(T) is the width of the zero-free region of the symmetric square L-function in Lemma 5. Using integration by parts and Lemma 3, we obtain where l is fulfilled apart from a fixed number of log-factors, We estimate S 1 first. The first and second derivatives of f 1 (σ) are Case 2. When x 9/20 ≤ T < x 4/5 . In this case, the function f 1 (σ) takes the extreme value at σ 1 , which gives max σ∈ [1/2,3/4] Combining all the above cases, we have Next, we need to estimate S 2 . It is easy to see that f 2 (σ) is linear function in [3/4, 1 − β], hence From (6), (7) and (8), we obtain which proves this lemma. (9) Proof. This lemma follows from (13) and (16) of Dufner [26].

Proof of Theorem 2
In this section, we write γ = 1/c. Then we have We split the summation range of q into two parts: q ≤ x c − 1 and x c − 1 < q ≤ x c to get For E 2 , there is at most one prime q x satisfying x c − 1 < q x ≤ x c , hence if we notice that |λ sym 2 f (p)| ≤ d 3 (p) 1. For E 1 , by the definition of the von Mangoldt function, we have The error term of the above formula contributes We use the same method to deal with the main term of (13) and get where the first O-term comes from By the range of n and Taylor's formula, we have log n = γ log m + O m −1 Then where the O-term comes from From (14), (15) and partial summation, we obtain where To get (3), it suffices to prove that The inequality (17) will be proved from a variant of (17) for short intervals. Let x 0 := 2, implies (17). We use Lemma 4 and get Taking where Under the Riemann Hypothesis, we have Making the change of variables u = log t, we deduce that Using the Cauchy-Schwarz inequality twice, we get Making the change of variables t = u(2π) −1 − (γ + log x v+1 + log x v )(4π) −1 , we deduce that the last integral in (22) can be written as where α = (γ + log x v+1 − log x v )(4π) −1 and Applying Lemma 1 with Θ = 1/2 and δ = Θ/α ∼ 2π/γ to estimate the integral in (23), we have From (20), (22) and (24), we get This combined with (16) and (17) gives Theorem 2 follows from (11), (12) and (25).

Proof of Theorem 1
To prove Theorem 1, we need Lemma 6 and Lemma 7. Furthermore, we only need to estimate Φ(γ, x v , T v ) unconditionally. Note that We consider the following integral mean value of Φ 1 (γ, x v , T), ε . From Lemma 6, we know that the upper bound of S depends on the range of T, so we have the following three cases. Case 1. When T ≥ x 4/5 v . In this case, we have x 1/γ 0 −δ 1 (ε) v < T and use (9) to get Case 2. When x 9/20 v ≤ T < x 4/5 v . In case of x 1/γ 0 −δ 1 (ε) v < T, we use (9) and get For convenience, we take t = log T log x v and consider a quadratic function g(t) = (2 + 2ε)t 2 − 2 √ 5t + 3 + 2/γ 0 .