Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a rigidity property for quadratic Gauss sums

Alexander P. Mangerel

TL;DR

The paper proves a rigidity phenomenon for quadratic Gauss sums: if a ±1-valued completely multiplicative function $f$ yields Gauss-sum–like dilation $S_f(ap) \approx g(p) S_f(a)$ on average over $a$, uniformly for primes $p \le N^c$, with $c>1/4$, then $f$ agrees with a real Dirichlet character modulo $N$ on almost all residue classes. The authors translate exponential-sum dilation bounds into mean-value statements, invoke Granville–Soundararajan theory to deduce a real character $\psi$ with strong correlation to $f$, and deduce that $f$ must coincide with $\psi$ off a sparse set. In the Liouville setting this yields a real zero near $s=1$ for $L(s,\chi)$, linking a Gauss-sum rigidity property to zero-distribution of Dirichlet L-functions. A conditional extension shows that, under a sufficiently wide zero-free region (Littlewood-type), the same conclusions hold for any $c>0$ and extend to arbitrary residue classes; this highlights a deep connection between exponential sums, character-like behavior, and L-function zeros with potential impact on understanding signs/factors in multiplicative functions.

Abstract

Let $N$ be a large prime and let $c > 1/4$. We prove that if $f$ is a $\pm 1$-valued completely multiplicative function, such that the exponential sums $$ S_f(a) := \sum_{1 \leq n < N} f(n) e(na/N), \quad a \pmod{N} $$ satisfy the ``Gauss sum-like'' approximate dilation symmetry property $$ \frac{1}{N}\sum_{a \pmod{N}} |S_f(ap) - f(p)S_f(a)|^2 = o(N), $$ uniformly over all primes $p \leq N^c$ then $f$ coincides with a real character modulo $N$ at all but $o(N)$ integers $1 \leq n < N$. As a consequence, taking $f$ to be the Liouville function we connect this exponential sums property to the location of real zeros of $L(s,χ)$ close to $s = 1$, for $χ$ the Legendre symbol modulo $N$. Assuming the $L$-functions of primitive Dirichlet characters modulo $N$ have a sufficiently wide zero-free region (of Littlewood type), we also show a more general result in which any $c > 0$ may be taken.

On a rigidity property for quadratic Gauss sums

TL;DR

The paper proves a rigidity phenomenon for quadratic Gauss sums: if a ±1-valued completely multiplicative function yields Gauss-sum–like dilation on average over , uniformly for primes , with , then agrees with a real Dirichlet character modulo on almost all residue classes. The authors translate exponential-sum dilation bounds into mean-value statements, invoke Granville–Soundararajan theory to deduce a real character with strong correlation to , and deduce that must coincide with off a sparse set. In the Liouville setting this yields a real zero near for , linking a Gauss-sum rigidity property to zero-distribution of Dirichlet L-functions. A conditional extension shows that, under a sufficiently wide zero-free region (Littlewood-type), the same conclusions hold for any and extend to arbitrary residue classes; this highlights a deep connection between exponential sums, character-like behavior, and L-function zeros with potential impact on understanding signs/factors in multiplicative functions.

Abstract

Let be a large prime and let . We prove that if is a -valued completely multiplicative function, such that the exponential sums satisfy the ``Gauss sum-like'' approximate dilation symmetry property uniformly over all primes then coincides with a real character modulo at all but integers . As a consequence, taking to be the Liouville function we connect this exponential sums property to the location of real zeros of close to , for the Legendre symbol modulo . Assuming the -functions of primitive Dirichlet characters modulo have a sufficiently wide zero-free region (of Littlewood type), we also show a more general result in which any may be taken.

Paper Structure

This paper contains 11 sections, 23 theorems, 167 equations.

Table of Contents

  1. Introduction and main results
  2. A conditional extension
  3. Remarks on the main results
  4. From primes to all residue classes $\pmod{N}$
  5. Unconditional results
  6. Conditional results
  7. Dilation by arbitrary residue classes
  8. Proof of Theorem \ref{['thm:dil']}
  9. From exponential sums to mean values
  10. Multiplicative functions that are close at a fixed scale
  11. Proofs of Theorem \ref{['thm:iff']} and Corollary \ref{['cor:Liou']}

Key Result

Theorem 1.1

Let $c > 1/4$. Let $N$ be a large prime, and let $f: \mathbb{N} \rightarrow \{-1,+1\}$ be multiplicative. Let $1 \leq M \leq N$, and suppose that for some $\pm 1$-valued sequence $(g(p))_{p \leq N^c}$. Then there is a real character $\psi$ modulo $N$ such that Moreover, there is an absolute constant $M_0$ such that if eq:unifSf holds with $M \geq M_0$ then and $g(p) = f(p)$ whenever $f(p) = \ps

Theorems & Definitions (49)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • ...and 39 more