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.
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The first author was supported by the Academy of Finland Grant No. 137883. The second author was partially supported by NSF Grant DMS-1128155.
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Matomäki, K., Radziwiłł, M. Sign changes of Hecke eigenvalues. Geom. Funct. Anal. 25, 1937–1955 (2015). https://doi.org/10.1007/s00039-015-0350-7
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DOI: https://doi.org/10.1007/s00039-015-0350-7