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 $x^{\delta}$ for some $\delta < 1$.