Oscillations and first-ever negative Fourier coefficients of symmetric square L-functions over sparse set

2025/10/21 by Srinivas Kotyada, Kotyada, Srinivas, Lalit Vaishya +1
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #Mathematical Analysis and Transform Methods #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2510.18994

openalex publication_date 2025/10/21 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

Let sym² f denote the symmetric square lift of a Hecke eigenform f ∈ S_k(Γ₀(N)) with the n^{\rm th}-Fourier coefficients λ_sym²f(n). In this article, we prove an estimate for the first moment of the sequence \ λ_sym²f(Q(\underlinex))\_Q ∈ S_D, \underlinex ∈ ℤ² where S_D denotes the set of in-equivalent reduced forms of the discriminant D. More precisely, we establish an estimate for the following sum: \beginsplit S(sym²f, D; X ) amp;= \sideset^\flat ∑_\substackQ(\underlinex) ≤ X
\underlinex ∈ ℤ² ,~ Q ∈ S_D
gcd(Q(\underlinex),N) =1 λ_sym²f(Q(\underlinex)), \endsplit Moreover, we consider a question concerning the behavior of signs of the Fourier coefficients λ_sym²f(n), supported on the set of integers represented by reduced forms of the discriminant D. We determine the size of n_{sym²f, D} (see definition before \thmrefExtMatKLSW), in terms of the conductor of the associated L-functions.

Citations

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

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