Signs of Fourier coefficients of cusp forms at integers represented by an integral binary quadratic form

Average estimates and sign change of Fourier coefficients of cusp forms at integers represented by binary quadratic form of fixed discriminant

“…Moreover, we use these estimates to obtain a result on sign change of sequence of the Fourier coefficients of the Hecke eigenforms supported at integers represented by a primitive integral binary quadratic form of fixed discriminant D < 0 with class number h(D) = 1. In this work, we also improve the result obtained in [ [1], [22]]. We refer to the introduction in [1], [22] for the history on the problem of sign change of the Fourier coefficients of cusp form.…”

“…In this work, we also improve the result obtained in [ [1], [22]]. We refer to the introduction in [1], [22] for the history on the problem of sign change of the Fourier coefficients of cusp form. Now, we fix some notations and state our results.…”

“…Remark 1.3. In [22], we obtain that for any fixed arbitrarily small constant ǫ > 0, this sequence changes its sign at least x 1 5 −ǫ many times in the interval (x, 2x] for sufficiently large x. In this work, we obtain an improved result.…”

“…We assume that the series and Euler product converge absolutely for ℜ(s) > 1 and L(s, F ) is an entire function except possibly for pole at s = 1 of order r and satisfies a nice functional equation (s → 1 − s). Then for any ǫ > 0, we have (22) uniformly for 0 ≤ σ ≤ 1. and |t| ≥ 1 where s = σ + it. Lemma 2.7.…”

Deterministic comparison of Hecke eigenvalues at the primes represented by a binary quadratic form

“…More precisely, they studied the distribution of {λ f (q(a, b))} where q(x, y) = x 2 + y 2 , and obtained an estimate for the summatory function k,l∈Z k 2 +l 2 ≤X λ f (q(k, l)). In our previous work (See [18,19]), we study the average behaviour of Hecke eigenvalue λ f (n), supported at the integers represented by primitive integral positive definite binary quadratic forms of fixed negative discriminant D., In a joint work with M.K. Pandey [20], we study the higher power moments of λ f (n), over the set of integers supported at the integers represented by primitive integral positive definite binary quadratic forms of fixed negative discriminant D. More precisely, we obtain an estimate for the sum (for each fixed r ∈ N and sufficiently large X ≥ 1)…”