First moment of Hecke eigenvalues at the integers represented by binary quadratic forms

2024/01/31 by Pandey, Manish Kumar, vaishya, Lalit · 1 citation
#11F11 #11M06 #FOS: Mathematics #Number Theory (math.NT) #Primary 11F30 #Secondary 11N37

paper · doi:10.48550/arxiv.2401.18055

Abstract

In the article, we consider a question concerning the estimation of summatory function of the Fourier coefficients of Hecke eigenforms indexed by a sparse set of integers. In particular, we provide an estimate for the following sum; \beginsplit S(f, Q; X ) amp;:= \sideset^\flat ∑_n= Q(\underlinex) ≤ X \atop gcd(n,N) =1 λ_f(n), \endsplit where \flat means that sum runs over the square-free positive integers, λ_f(n) denotes the normalised n^{\rm th} Fourier coefficients of a Hecke eigenform f of integral weight k for the congruence subgroup Γ₀(N) and Q is a primitive integral positive-definite binary quadratic forms of fixed discriminant D<0 with the class number h(D)=1. As a consequence, we determine the size, in terms of conductor of associated L-function, for the first sign change of Hecke eigenvalues indexed by the integers which are represented by Q. This work is an improvement and generalisation of the previous results.

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