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

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1. CMB Online first

Williams, Kenneth S.
Ternary quadratic forms and eta quotients
Let $\eta(z)$ $(z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0)$ denote the Dedekind eta function. We use a recent product-to-sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly 10 eta quotients \[ f(z):=\eta^{a(m_1)}(m_1 z)\cdots \eta^{{a(m_r)}}(m_r z)=\sum_{n=1}^{\infty}c(n)e^{2\pi i nz},\quad z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0, \] such that the Fourier coefficients $c(n)$ vanish for all positive integers $n$ in each of infinitely many non-overlapping arithmetic progressions. For example, it is shown that for $f(z)=\eta^4(z)\eta^{9}(4z)\eta^{-2}(8z)$ we have $c(n)=0$ for all $n$ in each of the arithmetic progressions $\{16k+14\}_{k \geq 0}$, $\{64k+56\}_{k \geq 0}$, $\{256k+224\}_{k \geq 0}$, $\{1024k+896\}_{k \geq 0}$, $\ldots$.

Keywords:Dedekind eta function, eta quotient, ternary quadratic forms, vanishing of Fourier coefficients, product-to-sum formula
Categories:11F20, 11E20, 11E25

2. CMB 2015 (vol 58 pp. 548)

Lü, Guangshi; Sankaranarayanan, Ayyadurai
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 formula
Categories:11F30, 11F66

3. 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 sequence
Categories:11F30, 11M41, 11B83

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