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Search: All articles in the CJM digital archive with keyword Fourier coefficients

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1. CJM 2011 (vol 63 pp. 634)

Lü, Guangshi
On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms
Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group. Let $\lambda_f(n)$ and $\lambda_g(n)$ be the $n$-th normalized Fourier coefficients of two holomorphic Hecke eigencuspforms $f(z), g(z) \in S_{k}(\Gamma)$, respectively. In this paper we are able to show the following results about higher moments of Fourier coefficients of holomorphic cusp forms.\newline (i) For any $\varepsilon>0$, we have \begin{equation*} \sum_{n\leq x}\lambda_f^5(n) \ll_{f,\varepsilon}x^{\frac{15}{16}+\varepsilon} \quad\text{and}\quad\sum_{n\leq x}\lambda_f^7(n) \ll_{f,\varepsilon}x^{\frac{63}{64}+\varepsilon}. \end{equation*} (ii) If $\operatorname{sym}^3\pi_f \ncong \operatorname{sym}^3\pi_g$, then for any $\varepsilon>0$, we have \begin{equation*} \sum_{n \leq x}\lambda_f^3(n)\lambda_g^3(n)\ll_{f,\varepsilon}x^{\frac{31}{32}+\varepsilon}; \end{equation*} If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$, then for any $\varepsilon>0$, we have \[ \sum_{n \leq x}\lambda_f^4(n)\lambda_g^2(n)=cx\log x+c'x+O_{f,\varepsilon}\bigl(x^{\frac{31}{32}+\varepsilon}\bigr); \] If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$ and $\operatorname{sym}^4\pi_f \ncong \operatorname{sym}^4\pi_g$, then for any $\varepsilon>0$, we have \[ \sum_{n \leq x}\lambda_f^4(n)\lambda_g^4(n)=xP(\log x)+O_{f,\varepsilon}\bigl(x^{\frac{127}{128}+\varepsilon}\bigr), \] where $P(x)$ is a polynomial of degree $3$.

Keywords: Fourier coefficients of cusp forms, symmetric power $L$-function
Categories:11F30, , , , 11F11, 11F66

2. CJM 2008 (vol 60 pp. 685)

Savu, Anamaria
Closed and Exact Functions in the Context of Ginzburg--Landau Models
For a general vector field we exhibit two Hilbert spaces, namely the space of so called \emph{closed functions} and the space of \emph{exact functions} and we calculate the codimension of the space of exact functions inside the larger space of closed functions. In particular we provide a new approach for the known cases: the Glauber field and the second-order Ginzburg--Landau field and for the case of the fourth-order Ginzburg--Landau field.

Keywords:Hermite polynomials, Fock space, Fourier coefficients, Fourier transform, group of symmetries
Categories:42B05, 81Q50, 42A16

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