TY - JOUR

T1 - Sign changes of Hecke eigenvalues

AU - Matomäki, Kaisa

AU - Radziwiłł, Maksym

N1 - Publisher Copyright:
© 2015, Springer International Publishing.

PY - 2015/12/1

Y1 - 2015/12/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00039-015-0350-7

DO - 10.1007/s00039-015-0350-7

M3 - Article

AN - SCOPUS:84949105680

SN - 1016-443X

VL - 25

SP - 1937

EP - 1955

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 6

ER -