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Search: MSC category 11F66 ( Langlands $L$-functions; one variable Dirichlet series and functional equations )

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1. CMB 2015 (vol 58 pp. 548)

 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 2015 (vol 58 pp. 580)

 A Specialisation of the Bump-Friedberg $L$-function We study the restriction of the Bump-Friedberg integrals to affine lines $\{(s+\alpha,2s),s\in\mathbb{C}\}$. It has a simple theory, very close to that of the Asai $L$-function. It is an integral representation of the product $L(s+\alpha,\pi)L(2s,\Lambda^2,\pi)$ which we denote by $L^{lin}(s,\pi,\alpha)$ for this abstract, when $\pi$ is a cuspidal automorphic representation of $GL(k,\mathbb{A})$ for $\mathbb{A}$ the adeles of a number field. When $k$ is even, we show that for a cuspidal automorphic representation $\pi$, the partial $L$-function $L^{lin,S}(s,\pi,\alpha)$ has a pole at $1/2$, if and only if $\pi$ admits a (twisted) global period, this gives a more direct proof of a theorem of Jacquet and Friedberg, asserting that $\pi$ has a twisted global period if and only if $L(\alpha+1/2,\pi)\neq 0$ and $L(1,\Lambda^2,\pi)=\infty$. When $k$ is odd, the partial $L$-function is holmorphic in a neighbourhood of $Re(s)\geq 1/2$ when $Re(\alpha)$ is $\geq 0$. Keywords:automorphic L functionsCategories:11F70, 11F66