关于第 j 次对称幂 L 函数在某些正整数稀疏序列上的傅立叶系数平均行为的说明

Pub Date : 2024-04-29 DOI:10.21136/cmj.2024.0038-24
Youjun Wang
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引用次数: 0

摘要

设 j ⩾ 2 为给定整数。让 Hk* 是全模态群 SL(2, ℤ)的偶数积分权重 k ⩾ 2 的所有归一化原始全形顶点形式的集合。对于 f∈ Hk*,用 \({{\rm{\lambda }}_{{\rm{sy}}{{\rm{m}}^j}{kern 1pt}} f}}(n)\) 表示连接到 f 的第 j 个对称幂 L 函数 (L(s, symjf)) 的第 n 个归一化傅里叶系数。我们感兴趣的是总和 $$\sum\limits_{\scriptstyle n\, = \、a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2x \atop \scriptstyle \,\,\,({a_1},{a_2},{a_3},{a_4},{a_5},{a_6}{\rm{)}}。\in\,{{mathbb{Z}}^6}}{{\rm{lambda }}_{{\rm{sy}}{{\rm{m}}^j}}\,f\left( n \right),}^2}$$ 其中 x 足够大,这改进了 A. Sharma 和 A. Sankaranarayanan (2023) 最近的工作。
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A note on average behaviour of the Fourier coefficients of jth symmetric power L-function over certain sparse sequence of positive integers

Let j ⩾ 2 be a given integer. Let Hk* be the set of all normalized primitive holomorphic cusp forms of even integral weight k ⩾ 2 for the full modulo group SL(2, ℤ). For fHk*, denote by \({{\rm{\lambda }}_{{\rm{sy}}{{\rm{m}}^j}{\kern 1pt} f}}(n)\) the nth normalized Fourier coefficient of jth symmetric power L-function (L(s, symjf)) attached to f. We are interested in the average behaviour of the sum

$$\sum\limits_{\scriptstyle n\, = \,a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2x \atop \scriptstyle \,\,\,\,\,\,\,({a_1},{a_2},{a_3},{a_4},{a_5},{a_6}{\rm{)}} \in \,{{\mathbb{Z}}^6}} {{\rm{\lambda }}_{{\rm{sy}}{{\rm{m}}^j}\,f\left( n \right),}^2}$$

where x is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).

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