Abstract
Let f and g be two distinct holomorphic cusp forms for S L 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 previous ones.
Highlights
Let H∗k be the set of all normalized Hecke primitive cusp forms of even integral weight k ≥ 2 for the full modular group SL2(Z) denoted by λf(n) the n-th Hecke eigenvalue of f ∈ H∗k . e Hecke eigenvalues of cusp forms have been extensively studied
There was a big breakthrough on the automorphy of all symmetric powers for cuspidal Hecke eigenforms, which implies that the L-function L(symjf, 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
Hp c1λfpj + c2λgpj − B (j + 1)2c1 +c22 +(j ac1λfpj + c2λgpj + 1)(|a| +|b|)c1 +c2
Summary
Let H∗k be the set of all normalized Hecke primitive cusp forms of even integral weight k ≥ 2 for the full modular group SL2(Z) denoted by λf(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]). 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|λf(pj)λg(pj) < 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 λf(p)λf(pj) for any even positive integer j. Inspired by [13], Chiriac [15] started to compare Hecke eigenvalues over prime numbers and simultaneously showed that the sets of primes for λf(p) < λg(p) and λ2f(p) < λ2g(p) both have analytic density at least 1/16. Notice that the pairSato–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]). This result is valid for the analytic density since the existence of the natural density implies that of the analytic density, and they are equal
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
