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$L^1$ means of exponential sums with multiplicative coefficients. II

Mayank Pandey, Maksym Radziwiłł

TL;DR

The paper proves a sharp rigidity phenomenon for real-valued, 1-bounded multiplicative functions: if the L^1 mass of the exponential sum $\sum_{n\le N} f(n) e(n\alpha)$ is sufficiently small, then on large primes $p$ the function $f(p)$ must approximate a quadratic Dirichlet character $\chi(p)$, with the conductor bounded in terms of the L^1 bound. The authors develop a uniform, major/minor arc framework, employ Gallagher's lemma and Mellin transform techniques to pass from additive to multiplicative structures, and prove a Halász-type large sieve bound controlling the major arcs via a pretentious distance $M_{f,\psi,T}$. Beyond the real-valued case, they extend to L^p norms and derive a general, uniform obstruction result: if $f$ is close to a non-quadratic character on primes, the pretentious distance cannot stay small; this yields uniform classification results and corollaries for automorphic L-functions. The work provides sharp, scale-uniform obstructions to additive structure in multiplicative sequences and offers new avenues for understanding when multiplicative coefficients mimic characters on primes.

Abstract

Let $f$ be a real-valued $1$-bounded multiplicative function. Suppose that the mean-value of $f^{2}$ exists, and $$\int_{0}^{1} \Big | \sum_{n \leq N} f(n)e^{2πi n α} \Big | d α\leq N^{o(1)}$$ as $N \rightarrow \infty$, then there exists a quadratic character $χ$ such that for every $δ> 0$ the (logarithmic) proportion of primes $p \leq N$ such that $|f(p) - χ(p)| < δ$ tends to $1$ as $N \rightarrow \infty$. More generally we show that for all $N, Δ\geq 1$ and $1$-bounded multiplicative functions $f$, if $$\int_{0}^{1} \Big | \sum_{n \leq N} f(n) e^{2πi n α} \Big | d α\leq Δ$$ and the $L^{2}$ norm of $f$ over $[1, N]$ is $\geq N / 100$, then $f$ pretends to be a multiplicative character of conductor $\leq Δ^{2}$ on primes in $[Δ^{2}, N]$. We highlight that the result is uniform in $f$, $N$ and $Δ$ and sharp as far as the size of the conductor goes. Moreover, the restriction to primes $p \in [Δ^{2}, N]$ turns out to be sharp in a suitably generalized version of this result, concerning sequences $f$ that are close $1\%$ of the time to multiplicative functions.

$L^1$ means of exponential sums with multiplicative coefficients. II

TL;DR

The paper proves a sharp rigidity phenomenon for real-valued, 1-bounded multiplicative functions: if the L^1 mass of the exponential sum is sufficiently small, then on large primes the function must approximate a quadratic Dirichlet character , with the conductor bounded in terms of the L^1 bound. The authors develop a uniform, major/minor arc framework, employ Gallagher's lemma and Mellin transform techniques to pass from additive to multiplicative structures, and prove a Halász-type large sieve bound controlling the major arcs via a pretentious distance . Beyond the real-valued case, they extend to L^p norms and derive a general, uniform obstruction result: if is close to a non-quadratic character on primes, the pretentious distance cannot stay small; this yields uniform classification results and corollaries for automorphic L-functions. The work provides sharp, scale-uniform obstructions to additive structure in multiplicative sequences and offers new avenues for understanding when multiplicative coefficients mimic characters on primes.

Abstract

Let be a real-valued -bounded multiplicative function. Suppose that the mean-value of exists, and as , then there exists a quadratic character such that for every the (logarithmic) proportion of primes such that tends to as . More generally we show that for all and -bounded multiplicative functions , if and the norm of over is , then pretends to be a multiplicative character of conductor on primes in . We highlight that the result is uniform in , and and sharp as far as the size of the conductor goes. Moreover, the restriction to primes turns out to be sharp in a suitably generalized version of this result, concerning sequences that are close of the time to multiplicative functions.
Paper Structure (14 sections, 20 theorems, 259 equations)

This paper contains 14 sections, 20 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. Generalizations and other consequences
  3. Notation
  4. Outline of the argument
  5. Orthogonality results
  6. Mean-value theorems
  7. Bounds for Mellin transforms
  8. Proof of Proposition A: The main criterion
  9. Proof of Lemma C: Minor arc bounds
  10. Proof of Lemma D: Reduction to completely multiplicative functions
  11. Proof of Proposition E: Bound for integral over major arcs
  12. Proof of Proposition G: A large sieve variant of Halasz's theorem
  13. Pretentious multiplicative functions
  14. Proof of Corollary \ref{['cor:first']}

Key Result

Corollary 1.1

Let $f : \mathbb{N} \rightarrow \mathbb{R}$ be a $1$-bounded multiplicative function such that, Suppose that there exists a $\psi(N) \rightarrow 0$ arbitrarily slowly with $N \rightarrow \infty$ such that, Then, there exists a quadratic Dirichlet character $\chi$ such that, as $N \rightarrow \infty$. That is for any $\delta > 0$, the proportion of $p$ (in a logarithmic sense) such that $|\chi(p

Theorems & Definitions (46)

  • Corollary 1.1
  • Remark 1
  • Remark 2
  • Corollary 1.2
  • proof
  • proof : Proof of Main Theorem
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 36 more