Conditional upper bounds on the least character non-residue

1. Problem Statement

The paper addresses the problem of bounding $n(\chi)$, the least character non-residue for a non-principal Dirichlet character $\chi$ modulo $q$.

• Definition: $n(\chi)$ is the smallest positive integer $n$ such that $(n, q)=1$ and $\chi(n) \neq 1$ (i.e., $\chi(n) \neq 0$).
• Context:
  • Vinogradov's Conjecture: For quadratic characters modulo a prime $p$, $n(p) \ll p^\epsilon$ for any $\epsilon > 0$.
  • Current Best Unconditional Bound: Burgess's bound gives $n(\chi) \ll q^{1/4\sqrt{e} + \epsilon}$ (for cube-free $q$).
  • Conditional Bounds: Assuming the Generalized Riemann Hypothesis (GRH), the bound improves significantly to $n(\chi) \ll (\log q)^2$.
• Goal: The author seeks to establish upper bounds on $n(\chi)$ that are conditional on specific zero-free regions within the critical strip of the associated $L$-function $L(s, \chi)$, particularly focusing on bounded rectangles along the line $\sigma = 1$ at arbitrary heights $t_0$.

2. Methodology

The paper employs analytic number theory techniques, specifically complex analysis of $L$-functions and inverse Mellin transforms.

• Core Technique: The proofs rely on analyzing weighted sums of the form $\sum w(n)\chi(n)\Lambda(n)$, where $\Lambda(n)$ is the von Mangoldt function.
• Contour Integration:
  • The author uses the inverse Mellin transform to express these sums as contour integrals involving the logarithmic derivative $L'/L(s, \chi)$.
  • The contour is shifted from the region of absolute convergence ($\sigma > 1$) to the left, picking up residues from poles and contributions from the zeros $\rho = \beta + i\gamma$ of $L(s, \chi)$.
• Kernel Construction (Theorem 3):
  • A specific kernel function $K(s, t_0)$ is constructed using a weight function $w(u) = (2\log y - |\log(x/u)|)(x/u)^{1+it_0}$.
  • This kernel is designed to isolate zeros near a specific height $t_0$ in the complex plane.
• Zero Density Estimates:
  • The proofs utilize standard bounds on the number of zeros in rectangles and discs (Lemmas 4 and 5, adapted from Montgomery and Vaughan).
  • The contribution of the zeros to the sum is bounded based on their distance from the point $1 + it_0$.

3. Key Contributions and Results

The paper presents two main theorems establishing conditional upper bounds based on the location of zeros.

Theorem 2: Global Strip Zero-Free Region

• Hypothesis: $L(s, \chi) \neq 0$ in the infinite strip $1-\delta < \sigma < 1$ (and its symmetric image), where $0 \le \delta < 1/2$.
• Result:
  $$n(\chi) \ll (\log q)^{1/\delta}$$
• Significance: This generalizes Ankeny's result. If the zero-free region is wider (larger $\delta$), the bound on the least non-residue becomes tighter (closer to $(\log q)^2$). If $\delta$ is small, the bound is weaker.

Theorem 3: Localized Rectangular Zero-Free Region

This is the paper's primary novel contribution, extending Rodosskii's work.

• Hypothesis: Let $|t_0| > 1$. Suppose $L(s, \chi) \neq 0$ in the region:
  $$1-\delta < \sigma < 1 \quad \text{and} \quad |t - t_0| \le K_1 |t_0|^2 \log q$$
  (i.e., a zero-free rectangle of width $\delta$ centered at height $t_0$).
• Result: There exist absolute constants $C, K_1$ such that:
  $$n(\chi) < C \left( \frac{K_1 |t_0|^2 \log q}{\delta} \right)^{1/\delta}$$
• Implication: This result allows one to "pinpoint" zeros. If the least non-residue is larger than this bound, it implies the existence of a zero within the specified rectangle near $t_0$.
• Special Case: When $t_0 = 0$, this recovers Rodosskii's bound with a zero-free region of smaller height.

Corollary/Remark 2

The author notes that by choosing $\delta$ as a function of $t_0$ (e.g., $\delta \approx 1/m$), one can derive bounds of the form $(\log q)^m$. This connects the size of the least non-residue directly to the "width" of the zero-free region at specific heights.

4. Significance and Future Directions

• Connection to Character Sums: The results bridge the gap between the size of character sums (and non-residues) and the distribution of zeros of $L$-functions. It refines the understanding of how "large" character sums imply the existence of zeros off the critical line.
• Refinement of GRH: While GRH assumes all zeros are on $\sigma=1/2$, this work quantifies the impact of zeros lying in specific sub-regions (strips or rectangles) near $\sigma=1$.
• Comparison to Previous Work:
  • It improves upon or generalizes results by Ankeny, Rodosskii, and Granville/Soundararajan.
  • It provides a more flexible framework than the global zero-free strip, allowing for localized analysis at arbitrary heights $t_0$.
• Future Directions:
  • The author proposes extending these techniques to sequences of zero-free regions where the width $\delta_i$ and height constraints $K_i$ vary with the imaginary part of the zeros.
  • This could lead to even stronger conditional bounds if one assumes a "Backlund-type" hypothesis where zeros are confined to narrower regions as $|t|$ increases.

Summary

Aritro Pathak's paper provides a rigorous analytic framework to derive upper bounds for the least character non-residue $n(\chi)$ contingent on the absence of zeros in specific rectangular regions near the line $\sigma=1$. By utilizing inverse Mellin transforms and carefully constructed kernel functions, the author demonstrates that the existence of a zero-free region of width $\delta$ at height $t_0$ forces $n(\chi)$ to be bounded by a power of $\log q$ dependent on $\delta$ and $t_0$. This work deepens the conditional understanding of the distribution of character non-residues and their relationship to the zero-density of $L$-functions.