Sign changes of Hecke eigenvalues

Abstract

Let f be a holomorphic or Maass Hecke cusp form for the full modular group and write ${\lambda_f(n)}$ for the corresponding Hecke eigenvalues. We are interested in the signs of those eigenvalues. In the holomorphic case, we show that for some positive constant ${\delta}$ and every large enough x, the sequence ${(\lambda_f(n))_{n \leq x}}$ has at least ${\delta x}$ sign changes. Furthermore we show that half of non-zero ${\lambda_f(n)}$ are positive and half are negative. In the Maass case, it is not yet known that the coefficients are non-lacunary, but our method is robust enough to show that on the relative set of non-zero coefficients there is a positive proportion of sign changes. In both cases previous lower bounds for the number of sign changes were of the form x ^δ for some δ < 1.