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Search: MSC category 11F30 ( Fourier coefficients of automorphic forms )

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1. CMB Online first

 Higher Moments of Fourier Coefficients of Cusp Forms Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group $SL(2, \mathbb{Z})$. Let $\lambda_f(n)$, $\lambda_g(n)$, $\lambda_h(n)$ be the $n$th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms $f(z) \in S_{k_1}(\Gamma), g(z) \in S_{k_2}(\Gamma), h(z) \in S_{k_3}(\Gamma)$ respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as $\lambda_f(n)^4\lambda_g(n)^2$, $\lambda_g(n)^6$, $\lambda_g(n)^2\lambda_h(n)^4$, and $\lambda_g(n^3)^2$ twisted by the arithmetic function $\lambda_f(n)$. Keywords:Fourier coefficients of automorphic forms, Dirichlet series, triple product $L$-function, Perron's formulaCategories:11F30, 11F66

2. CMB 2011 (vol 54 pp. 757)

Sun, Qingfeng
 Cancellation of Cusp Forms Coefficients over Beatty Sequences on $\textrm{GL}(m)$ Let $A(n_1,n_2,\dots,n_{m-1})$ be the normalized Fourier coefficients of a Maass cusp form on $\textrm{GL}(m)$. In this paper, we study the cancellation of $A (n_1,n_2,\dots,n_{m-1})$ over Beatty sequences. Keywords:Fourier coefficients, Maass cusp form on $\textrm{GL}(m)$, Beatty sequenceCategories:11F30, 11M41, 11B83

3. CMB 2009 (vol 52 pp. 195)

Garaev, M. Z.; Garcia, V. C.; Konyagin, S. V.
 The Waring Problem with the Ramanujan $\tau$-Function, II Let $\tau(n)$ be the Ramanujan $\tau$-function. We prove that for any integer $N$ with $|N|\ge 2$ the diophantine equation $$\sum_{i=1}^{148000}\tau(n_i)=N$$ has a solution in positive integers $n_1, n_2,\ldots, n_{148000}$ satisfying the condition $$\max_{1\le i\le 148000}n_i\ll |N|^{2/11}e^{-c\log |N|/\log\log |N|},$$ for some absolute constant $c>0.$ Categories:11B13, 11F30

4. CMB 2002 (vol 45 pp. 364)

Deitmar, Anton
 Mellin Transforms of Whittaker Functions In this note we show that for an arbitrary reductive Lie group and any admissible irreducible Banach representation the Mellin transforms of Whittaker functions extend to meromorphic functions. We locate the possible poles and show that they always lie along translates of walls of Weyl chambers. Categories:11F30, 22E30, 11F70, 22E45