Ω-RESULT ON COEFFICIENTS OF AUTOMORPHIC L-FUNCTIONS OVER SPARSE SEQUENCES
Ω-RESULT ON COEFFICIENTS OF AUTOMORPHIC L-FUNCTIONS OVER SPARSE SEQUENCES
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DOI：
10.4134/jkms.2015.52.5.945
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发表时间：
2015-09
影响因子：
4.5
通讯作者：
H. Lao;H. Wei
中科院分区：
文献类型：
--
作者：
H. Lao;H. Wei
Abstract. Let λ f (n) denote the n-th normalized Fourier coefficient ofa primitive holomorphic form f for the full modular group Γ = SL 2 (Z).In this paper, we are concerned with Ω-result on the summatorPy function n6x λ 2f (n 2 ), and establish the following resultX n6x λ 2f (n 2 ) = c 1 x + Ω(x 49 ),where c 1 is a suitable constant. 1. Introduction and main resultsAccording to the Langlands program, there are many hidden structures un-derlying the Fourier coefficients of an automorphic form. Thus it is very impor-tant and essential to investigateits summatoryfunction overa certain sequence.Let H ∗ k be the set of all normalized Hecke eigencuspforms of even integralweight kfor the full modular group Γ = SL 2 (Z). For f(z) ∈ H ∗ k , f(z) has thefollowing Fourier expansion at the cusp ∞f(z) =X ∞n=1 λ f (n)n k−12 e 2πinz ,where λ f (n) is real and satisfies the multiplicative property(1.1) λ f (m)λ f (n) =X d|(m,n) λ f mnd 2 for any integers m≥ 1 and n≥ 1.The size and oscillations of λ f (n) deserve deep research. In 1974, Deligne
