Aperiodicity of signs of Hecke eigenvalues

  • Guangshi LüEmail author


Let f be a normalized Hecke eigenform of weight k and arbitrary level, and \(\lambda _f(n)\) be the nth normalized Fourier coefficient of f. Let g(n) be any multiplicative function of modulus at most 1, which satisfies that for all primes p, g(p) are all positive (or are all negative). It is derived that for each \(\ell \in {\mathbb {N}}\), and every Dirichlet character \(\chi \), \(g(n)sgn(\lambda _f(n^{\ell }))\), \(g(n)\chi (n)sgn(\lambda _f(n^{\ell }))\) and other related functions are (strongly) aperiodic multiplicative functions. This gives that they are \(U^s\)-uniform and then fully oscillating sequences. As corollaries, they are orthogonal to any dynamical sequence arising from topological dynamical systems with quasi-discrete spectrum, and multiple ergodic realizations of affine maps of zero topological entropy on compact Abelian groups. We further find that they satisfy an averaged version of a conjecture of Elliott regarding correlations of bounded multiplicative functions.


Multiplicative function Hecke eigenvalue Aperiodicity Oscillating sequences 

Mathematics Subject Classification

11N37 11F30 11F11 



This work is supported in part by NSFC (Nos. 11771252, 11531008), IRT16R43, and Taishan Scholars Project. The author would like to thank the reviewer for valuable suggestions and detailed comments, and to thank Dr. X. He for useful discussion.


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© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanChina

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