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Published December 2015 | Submitted
Journal Article Open

Sign changes of Hecke eigenvalues


Let f be a holomorphic or Maass Hecke cusp form for the full modular group and write λ_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 δ and every large enough x, the sequence (λ_f(n))_(n≤x) has at least δx sign changes. Furthermore we show that half of non-zero λ_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.

Additional Information

© 2015 Springer International Publishing. Received: 11 August 2014; Revised: 17 April 2015; Accepted: 21 April 2015; First Online: 25 November 2015. The first author was supported by the Academy of Finland Grant No. 137883. The second author was partially supported by NSF Grant DMS-1128155. The authors would like to thank the anonymous referee for suggesting a complete re-write of the proof (a variant of which is presented here). The referee's argument led to several improvements and to a much cleaner proof, for which the authors are extremely grateful. The authors would also like to thank Andrew Granville for providing the proof of Lemma 2.1 and Matti Jutila for mentioning the shifted convolution problem. They are also grateful to Jie Wu and Wenguang Zhai for an e-mail discussion concerning [WZ13] and to Eeva Vehkalahti for helpful comments on earlier versions of this manuscript. They also would like to thank Peter Sarnak for a lemma which was used in the previous version of the paper and for posing the question on the entropy of λ_f(n). Finally the authors are grateful to Brian Conrey for suggesting the use of a mollifier at the October 2013 meeting in Oberwolfach.

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