On an open problem of Sankaranarayanan  

作　　者：L GuangShi School of Mathematics,Shandong University,Jinan 250100,China 

出　　处：《Science China Mathematics》2010年第5期282-287,共6页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No.10701048);Natural Science Foundation of Shandong Province (Grant No.ZR2009AM007) and IIFSDU

摘　　要：Let λ(n) be the n-th normalized Fourier coeficient of a holomorphic Hecke eigenform f(z) ∈ Sk(Γ) for the full modular group.In one of his papers,Sankaranarayanan mentioned that it is an open problem to give a non-trivial estimate for the sum of λ(n) over cubes,i.e.n x λ(n3).In this paper,we are able to use the analytic properties of symmetric power L-functions to solve his problem.More precisely,we prove that Σn≤zλ(n3)【【 x(3/4 +ε),Σn≤zλ(n4)【【 x(97+ε).Let λ(n) be the n-th normalized Fourier coeficient of a holomorphic Hecke eigenform f(z) ∈ Sk(Γ) for the full modular group.In one of his papers,Sankaranarayanan mentioned that it is an open problem to give a non-trivial estimate for the sum of λ(n) over cubes,i.e.n x λ(n3).In this paper,we are able to use the analytic properties of symmetric power L-functions to solve his problem.More precisely,we prove that Σn≤zλ(n3)<< x(3/4 +ε),Σn≤zλ(n4)<< x(97+ε).

关 键 词：FOURIER coeficients of CUSP FORMS SYMMETRIC power L-FUNCTION LANGLANDS program 

分 类 号：O174.2[理学—数学]