A short note on sign changes

Abstract

In this paper, we present a quantitative result for the number of sign changes for the sequences {a(n j )} n≥1 , j = 2, 3, 4 of the Fourier coefficients of normalized Hecke eigen cusp forms for the full modular group SL 2 (Z). We also prove a similar kind of quantitative result for the number of sign changes of the q-exponents c(p) (p vary over primes) of certain generalized modular functions for the congruence subgroup Γ 0 (N ), where N is square-free.

Sign changes of Hecke eigenvalues

“…This motivates the point of view that the signs of λ f (n) are GL(2) analogues of real characters. The frequency of signs and sign changes and other related questions have been also recently studied in [1,15,16,18,23]. In [5] Ghosh and Sarnak relate sign changes of λ f (n) to zeros of f (z) on the imaginary line.…”

Sign changes of coefficients of certain Dirichlet series

“…Remark 12. In [16], certain quantitative results for the sign change in each subsequence {a n j } ∞ n=1 , j ∈ {2, 3, 4} have been established and hence infinitely many sign change in the subsequences. The proof uses suitable bounds for the two average sums 1≤n≤x a(n j ) and 1≤n≤x a(n j ) 2 .…”

“…Using these results, we conclude the oscillatory behaviour (infinitely many sign changes) of the Fourier coefficients of elliptic cusp forms, Siegel cusp forms, Maass cusp forms and second order cusp forms. The sign change results in the case of elliptic cusp forms, Siegel cusp forms and symmetric power L-functions associated to elliptic cusp forms have been established in [8,9,16] using different techniques. The applications provided in the case of second order cusp forms and Maass cusp forms are new.…”

A note on signs of Fourier coefficients of two cusp forms

“…al. [15] investigated that for a normalized Hecke eigen cusp form with Fourier coefficients {a(n)}, the subsequences {a(n j )} n≥1 (j = 2, 3, 4) of the Fourier coefficients change signs infinitely often. W.Kohnen and Y.Martin [9] generalized their result by showing the subsequence {a(p jn )} n≥0 have infinitely many sign changes for almost all primes p and j ∈ N .…”