Sign changes of Hecke eigenvalues

Kaisa Matomäki, Maksym Radziwiłł*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

Abstract

Let f be a holomorphic or Maass Hecke cusp form for the full modular group and write λf(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 δ and every large enough x, the sequence (Formula presented.) has at least δx sign changes. Furthermore we show that half of non-zero λf(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 xδ for some δ < 1.

Original languageEnglish (US)
Pages (from-to)1937-1955
Number of pages19
JournalGeometric and Functional Analysis
Volume25
Issue number6
DOIs
StatePublished - Dec 1 2015

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

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