Density results on the coefficients of the triple product $ L $-functions

Ying Han; Huixue Lao; Ying Han; Huixue Lao

doi:10.3934/math.2026059

AIMS Mathematics

2026, Volume 11, Issue 1: 1382-1411. doi: 10.3934/math.2026059

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Density results on the coefficients of the triple product $ L $-functions

Ying Han ,
Huixue Lao ^,

School of Mathematics and Statistics, Shandong Normal University, Ji'nan, 250358, China

Received: 11 September 2025 Revised: 17 December 2025 Accepted: 23 December 2025 Published: 16 January 2026
MSC : 11F30, 11F11, 11F66

Let $ H_{k}^{\ast} $ denote the set of normalized primitive holomorphic Hecke cusp forms of even integral weight $ k $ for the full modular group. Denote by $ \lambda_{f\times f\times f}(n) $ the $ n $th coefficient of the triple product $ L $-function $ L(f\times f\times f, s) $ attached to $ f\in H_{k}^{\ast} $. Suppose $ Q(\underline x) $ is a primitive integral positive-definite binary quadratic form of fixed discriminant $ D < 0 $ with the class number $ h(D) = 1 $. In this paper, we study the distribution of $ \lambda_{f\times f\times f}(n) $ on the set of all primes and its subset $ \{p:p = Q(\underline x)\} $ and obtain the analytic density and the natural density of the above sets. These results generalize previous ones.
- binary quadratic form,
- triple product $ L $-function,
- analytic density,
- natural density
Citation: Ying Han, Huixue Lao. Density results on the coefficients of the triple product $ L $-functions[J]. AIMS Mathematics, 2026, 11(1): 1382-1411. doi: 10.3934/math.2026059

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Abstract

Let $ H_{k}^{\ast} $ denote the set of normalized primitive holomorphic Hecke cusp forms of even integral weight $ k $ for the full modular group. Denote by $ \lambda_{f\times f\times f}(n) $ the $ n $th coefficient of the triple product $ L $-function $ L(f\times f\times f, s) $ attached to $ f\in H_{k}^{\ast} $. Suppose $ Q(\underline x) $ is a primitive integral positive-definite binary quadratic form of fixed discriminant $ D < 0 $ with the class number $ h(D) = 1 $. In this paper, we study the distribution of $ \lambda_{f\times f\times f}(n) $ on the set of all primes and its subset $ \{p:p = Q(\underline x)\} $ and obtain the analytic density and the natural density of the above sets. These results generalize previous ones.

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