Letλf(n)be the normalized n-th Fourier coefficient of holomorphic eigenform f for the full modular group and Pc(x):={p≤x|[pc]prime},c∈R+.In this paper,we show that for all 0<c<1 the mean value ofλf(n)in Pc(x...Letλf(n)be the normalized n-th Fourier coefficient of holomorphic eigenform f for the full modular group and Pc(x):={p≤x|[pc]prime},c∈R+.In this paper,we show that for all 0<c<1 the mean value ofλf(n)in Pc(x)is x log-A x assuming the Riemann Hypothesis.Unconditionally,in the sense of Lebesgue measure,it holds for almost all c∈(ε,1-ε).展开更多
Letπbe a self-dual irreducible cuspidal automorphic representation of GL_(2)(A_(Q))with trivial central character.Its Hecke eigenvalue λπ(n)is a real multiplicative function in n.We show that λπ(n)<0 for some ...Letπbe a self-dual irreducible cuspidal automorphic representation of GL_(2)(A_(Q))with trivial central character.Its Hecke eigenvalue λπ(n)is a real multiplicative function in n.We show that λπ(n)<0 for some n<<Q^(2/5)_(π),where Qπdenotes(a special value of)the analytic conductor.The value 2/5 is the first explicit exponent for Hecke-Maass newforms.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.11771256,11971476)。
文摘Letλf(n)be the normalized n-th Fourier coefficient of holomorphic eigenform f for the full modular group and Pc(x):={p≤x|[pc]prime},c∈R+.In this paper,we show that for all 0<c<1 the mean value ofλf(n)in Pc(x)is x log-A x assuming the Riemann Hypothesis.Unconditionally,in the sense of Lebesgue measure,it holds for almost all c∈(ε,1-ε).
基金supported by General Research Fund of the Research Grants Council of Hong Kong(Grant Nos.17313616 and 17305617)supported by National Natural Science Foundation of China(Grant No.11871193)+1 种基金the Program for Young Scholar of Henan Province(Grant No.2019GGJS026)supported by National Natural Science Foundation of China(Grant No.11871344)。
文摘Letπbe a self-dual irreducible cuspidal automorphic representation of GL_(2)(A_(Q))with trivial central character.Its Hecke eigenvalue λπ(n)is a real multiplicative function in n.We show that λπ(n)<0 for some n<<Q^(2/5)_(π),where Qπdenotes(a special value of)the analytic conductor.The value 2/5 is the first explicit exponent for Hecke-Maass newforms.
