On density of the zeros of Dedekind zeta-functions

1. Problem Statement

The paper investigates the zero-density problem for the Dedekind zeta function $\zeta_K(s)$ associated with an algebraic number field $K$ of degree $k = [K:\mathbb{Q}]$.

• Definition: Let $N_\zeta(\sigma, K, T)$ denote the number of zeros $\rho = \beta + i\gamma$ of $\zeta_K(s)$ such that $|\gamma| \le T$ and $\beta \ge \sigma$, counted with multiplicity.
• Goal: To establish upper bounds for $N_\zeta(\sigma, K, T)$ in the critical strip $1/2 \le \sigma \le 1$ as $T \to \infty$.
• Context:
  • For $k=2$ (quadratic fields), previous results by Heath-Brown provided bounds with exponents depending on $\sigma$.
  • For $k \ge 3$, Heath-Brown (1977) established $N_\zeta(\sigma, K, T) \ll T^{k(1-\sigma)+\epsilon}$. Later, using subconvexity bounds, this was improved to $T^{((6+2k)/3)(1-\sigma)+\epsilon}$.
  • The paper aims to improve these bounds, particularly for the range $5/6 \le \sigma \le 1$ when $3 \le k \le 6$, and for $k \ge 7$.

2. Methodology

The author employs a combination of classical analytic number theory techniques and recent advances in mean value estimates. The core methodology involves:

A. Zero-Detecting Technique

The proof utilizes the standard "zero-detecting" method. The author considers the product $\zeta_K(s) M_X(s, K)$, where $M_X(s, K)$ is a truncated Dirichlet series of the Möbius function $\mu_K(n)$.

• If $\zeta_K(s)$ has a zero $\rho$, then the truncated series must be small, or the tail of the series must be large.
• This leads to a dichotomy: a zero $\rho$ must satisfy either a large sum condition involving the coefficients $D_X(n, K)$ (Class 1 zeros) or a large integral condition involving $\zeta_K$ on a vertical line (Class 2 zeros).

B. Separation of Zeros

The total count $N_\zeta$ is bounded by $R_1 + R_2$ (plus logarithmic factors), where:

• $R_1$: Zeros satisfying the discrete sum condition (large values of the Dirichlet polynomial).
• $R_2$: Zeros satisfying the integral condition (large values of $\zeta_K$ on the line $\Re(s) = 1/2$).

C. Key Analytic Tools

1. Subconvexity Bounds (Lemma 2.1): The proof relies heavily on the recent result by [5] which establishes the subconvexity bound for all number fields (abelian or not):
   $$\zeta_K(1/2 + it) \ll t^{k/6 + \epsilon}$$
   This is a crucial improvement over the classical bound $t^{k/4}$.
2. Exponent Pairs: The estimation of $R_1$ utilizes exponent pairs $(\kappa, \lambda)$. The author applies Huxley's subdivision method to split the interval $[-T, T]$ into smaller subintervals of length $T_0$.
3. Halász-Montgomery Inequality: To bound the number of zeros satisfying the large sum condition, the author uses the Halász-Montgomery inequality to estimate the mean square of the Dirichlet polynomial.
4. Mellin Transform: Used to estimate the off-diagonal terms in the mean value calculation, specifically the function $G(it) = \sum e^{-n/N} n^{-it}$.

3. Key Contributions and Results

The paper presents two main theorems improving upon previous results by Heath-Brown and others.

Theorem 1.1: The Case $k=2$ (Quadratic Fields)

For quadratic fields, the author derives a sharper bound using an arbitrary exponent pair $(\kappa, \lambda)$.

• Result:
  $$N_\zeta(\sigma, K, T) \ll T^{\frac{4}{6\sigma - 3}(1-\sigma) + \epsilon}$$
• Condition: Valid for $\sigma$ in the range defined by the exponent pair:
  $$\frac{1 + \lambda - 6\kappa}{2 - 8\kappa} \le \sigma < 1$$
  (with constraints $\kappa < 1/4$ and $\kappa + 1 \le 5\lambda/2$).
• Specific Improvement: By choosing the exponent pair $(\kappa, \lambda) = (1/14, 11/14)$, the bound holds for $19/20 < \sigma \le 1$. This improves the range and the exponent compared to Heath-Brown's results for $k=2$.

Theorem 1.2: The Case $k \ge 3$ (General Number Fields)

This is the primary contribution for higher-degree fields.

• Result:
  $$N_\zeta(\sigma, K, T) \ll T^{\frac{2k}{6\sigma - 3}(1-\sigma) + \epsilon}$$
• Range: Valid for
  $$\frac{2k + 3}{2k + 6} \le \sigma < 1$$
• Comparison with Previous Work:
  • For $3 \le k \le 6$: The exponent $\frac{2k}{6\sigma - 3}$ is strictly smaller than Heath-Brown's $k$ (for $\sigma \ge 5/6$), providing a significant improvement.
  • For $k \ge 7$: The result improves upon the subconvexity-based bound $T^{((6+2k)/3)(1-\sigma)}$ because $\frac{2k}{6\sigma - 3} < \frac{6+2k}{3}$ in the specified range.

4. Significance and Impact

1. Refinement of Zero Density Estimates: The paper successfully tightens the upper bounds for the number of zeros of Dedekind zeta functions in the critical strip. This is a fundamental problem in analytic number theory, closely related to the Generalized Riemann Hypothesis (GRH).
2. Utilization of Subconvexity: The work demonstrates how the recent breakthrough in subconvexity bounds ($\zeta_K(1/2+it) \ll t^{k/6+\epsilon}$) can be effectively integrated into zero-density estimates to yield stronger results than previously possible using older bounds ($t^{k/4}$).
3. Extension to Higher Degrees: While much of the literature focuses on $k=2$ or general $k$ with weaker bounds, this paper provides specific, optimized estimates for $k \ge 3$, covering the critical range $5/6 \le \sigma \le 1$ where the density of zeros is most relevant for applications (e.g., in the study of prime ideals).
4. Methodological Synthesis: The paper effectively synthesizes Huxley's subdivision technique, exponent pair theory, and modern mean value estimates to handle the specific arithmetic structure of Dedekind zeta functions.

In summary, Wei Zhang provides a rigorous improvement on the zero-density estimates for Dedekind zeta functions, leveraging recent subconvexity results to achieve sharper exponents for both quadratic and higher-degree number fields.