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Sums of Fourier coefficients involving theta series and Dirichlet characters

Yanxue Yu

Abstract

Let $f$ be a holomorphic or Maass cusp forms for $ \rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $λ_f(n)$ and \bna r_{\ell}(n)=\#\left\{(n_1,\cdots,n_{\ell})\in \mathbb{Z}^2:n_1^2+\cdots+n_{\ell}^2=n\right\}. \ena Let $χ$ be a primitive Dirichlet character of modulus $p$, a prime. In this paper, we are concerned with obtaining nontrivial estimates for the sum \bna \sum_{n\geq1}λ_f(n)r_{\ell}(n)χ(n)w\left(\frac{n}{X}\right) \ena for any $\ell \geq 3$, where $w(x)$ be a smooth function compactly supported in $[1/2,1]$.

Sums of Fourier coefficients involving theta series and Dirichlet characters

Abstract

Let be a holomorphic or Maass cusp forms for with normalized Fourier coefficients and \bna r_{\ell}(n)=\#\left\{(n_1,\cdots,n_{\ell})\in \mathbb{Z}^2:n_1^2+\cdots+n_{\ell}^2=n\right\}. \ena Let be a primitive Dirichlet character of modulus , a prime. In this paper, we are concerned with obtaining nontrivial estimates for the sum \bna \sum_{n\geq1}λ_f(n)r_{\ell}(n)χ(n)w\left(\frac{n}{X}\right) \ena for any , where be a smooth function compactly supported in .

Paper Structure

This paper contains 7 sections, 8 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Proof of Theorem \ref{['Th11']}
  4. The minor arcs
  5. The major arcs
  6. Conclusion
  7. Proof of Theorem \ref{['Th12']}

Key Result

Theorem 1.1

Let $f\in H_\kappa$ or $S_\mu$ and $\chi$ be a primitive Dirichlet character of modulus $p$, a prime. Let $w:\mathbb{R}\to[0,\infty)$ be a smooth function compactly supported in $[1/2, 1]$ and satisfying where $1\leq\Delta<X$. Then for any $\ell \geq 3$, we have for $p< X$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: Hua's lemma
  • proof
  • Proposition 3.1
  • ...and 3 more