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Variance of GL(2) Fourier coefficients in arithmetic progressions

Laurent Montaigu

Abstract

We improve a result of Lau and Zhao on the variance of Fourier coefficients of primitive cuspidal modular forms for SL2(Z) in arithmetic progressions. This is achieved by using bounds on the first moment of Rankin-Selberg L-functions in the height aspect and non-trivial estimates for shifted convolution sums.

Variance of GL(2) Fourier coefficients in arithmetic progressions

Abstract

We improve a result of Lau and Zhao on the variance of Fourier coefficients of primitive cuspidal modular forms for SL2(Z) in arithmetic progressions. This is achieved by using bounds on the first moment of Rankin-Selberg L-functions in the height aspect and non-trivial estimates for shifted convolution sums.
Paper Structure (16 sections, 33 theorems, 325 equations)

This paper contains 16 sections, 33 theorems, 325 equations.

Table of Contents

  1. Introduction
  2. Variances of arithmetic functions
  3. Statement of the main results
  4. Preliminaries and background
  5. Standard analytic and modular tools
  6. Special function in the Voronoi summation formula
  7. Variance of Fourier coefficients in arithmetic progressions
  8. Smoothing out the sum
  9. Main Term
  10. Error term
  11. Final computations
  12. Variance of the divisor function in arithmetic progressions
  13. Main Term
  14. Error terms
  15. Fake main terms
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assume $X^{\frac{1}{2}}<q<X$, then for any $\varepsilon>0$ where $d_{\frac{1}{2}}(q)$ is the smallest divisor of $q$ greater than $X^{\frac{1}{2}}$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.2
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • ...and 37 more