Some Density Results on Sets of Primes for Hecke Eigenvalues

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.