On the average behavior of the Fourier coefficients of j^th symmetric power L-function over a certain sequences of positive integers

2022/06/03 by Anubhav Sharma, Sharma, Anubhav, Ayyadurai Sankaranarayanan +1 · 1 citation
Mathematics · #11F11 #11F30 #11M06 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2206.01491

openalex publication_date 2022/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate the average behavior of the n^th normalized Fourier coefficients of the j^th (j ≥ 2 be any fixed integer) symmetric power L-function (i.e., L(s,sym^jf)), attached to a primitive holomorphic cusp form f of weight k for the full modular group SL(2,ℤ) over a certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum ∑_\stackrela₁²+a₂²+a₃²+a₄²+a₅²+a₆²≤ x(a₁,a₂,a₃,a₄,a₅,a₆)∈ℤ⁶λ²_sym^jf(a₁²+a₂²+a₃²+a₄²+a₅²+a₆²), where x is sufficiently large, and L(s,sym^jf):=∑_n=1^∞\dfracλ_sym^jf(n)n^s. When j=2, the error term which we obtain, improves the earlier known result.

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