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Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$

Alexandru Pascadi

Abstract

We prove new large sieve inequalities for the Fourier coefficients $ρ_{j\mathfrak{a}}(n)$ of exceptional Maass forms of a given level, weighted by sequences $(a_n)$ with sparse Fourier transforms - including two key types of sequences that arise in the dispersion method. These give the first savings in the exceptional spectrum for the critical case of sequences as long as the level, and lead to improved bounds for various multilinear forms of Kloosterman sums. As an application, we show that the greatest prime factor of $n^2+1$ is infinitely often greater than $n^{1.3}$, improving Merikoski's previous threshold of $n^{1.279}$. We also announce applications to the exponents of distribution of primes and smooth numbers in arithmetic progressions.

Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$

Abstract

We prove new large sieve inequalities for the Fourier coefficients of exceptional Maass forms of a given level, weighted by sequences with sparse Fourier transforms - including two key types of sequences that arise in the dispersion method. These give the first savings in the exceptional spectrum for the critical case of sequences as long as the level, and lead to improved bounds for various multilinear forms of Kloosterman sums. As an application, we show that the greatest prime factor of is infinitely often greater than , improving Merikoski's previous threshold of . We also announce applications to the exponents of distribution of primes and smooth numbers in arithmetic progressions.
Paper Structure (23 sections, 30 theorems, 287 equations, 2 figures)

This paper contains 23 sections, 30 theorems, 287 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The large sieve inequalities
  3. Acknowledgements
  4. Informal overview
  5. Large sieve with general sequences
  6. Exponential phases and a counting problem
  7. Sequences with frequency concentration
  8. Multilinear forms of Kloosterman sums
  9. Layout of paper
  10. Notation and preliminaries
  11. Standard analytic notation
  12. Cusps, automorphic forms, Kloosterman sums
  13. The Kuznetsov formula and exceptional eigenvalues
  14. Bounds for Fourier coefficients
  15. Combinatorial bounds
  16. ...and 8 more sections

Key Result

Theorem 1

For infinitely many $n \in \mathbb{Z}_+$, the greatest prime factor of $n^2+1$ is larger than $n^{1.3}$.

Figures (2)

  • Figure 1: Structure of paper (arrows signify logical implications).
  • Figure 2: Type I (left) and Type II (right) ranges. Previous results in gray; our improvements in blue; conditional ranges in red (assuming Selberg's eigenvalue conjecture).

Theorems & Definitions (81)

  • Theorem 1
  • Theorem 1: Large sieve with general sequences deshouillers1982kloosterman
  • Remark
  • Remark
  • Theorem 2: Large sieve with exponential phases
  • Remark
  • Theorem 3: Large sieve with dispersion coefficients
  • Remark
  • Remark
  • Remark
  • ...and 71 more