On the Local Maxima Behaviour of Hecke Eigenvalues and Its Applications

Abstract

Let Hk denote the space of primitive holomorphic cusp forms of even integral weight k for the full modular group Γ = SL(2, ℤ). Denote by $${\lambda _{{\rm{sy}}{{\rm{m}}^m}f}}(n)$$ the nth normalized coefficient of the Dirichlet expansion of the mth symmetric power L-function associated to f. In this paper, we establish the asymptotic formulas of sums of pairwise maxima concerning the normalized coefficients of symmetric power L-functions. We also establish similar results for the normalized coefficients of Rankin-Selberg L-functions L(symif × symjf, s) and L(symif × symjg, s) attached to f and g, respecitvely. As applications, we also consider the proportion of the sign changes among the difference of normalized coefficients associated with symmetric power L-functions and Rankin-Selberg L-functions attached to f and g, respectively.