Sharp Bounds for Moments of the Dedekind Zeta Function

Technical Summary: Sharp Bounds for Moments of the Dedekind Zeta Function

Problem Statement
The central problem addressed in this paper is the establishment of sharp upper bounds for the moments of the Dedekind zeta function, $\zeta_K(s)$, associated with a finite Galois extension $K/\mathbb{Q}$, on the critical line $\Re(s) = 1/2$. Specifically, the authors investigate the shifted moments:
$$\int_0^T \prod_{j=1}^{2k} \left| \zeta_K\left(\frac{1}{2} + i(t + b_j)\right) \right|^{a_j} dt$$
under the assumption of the Generalized Riemann Hypothesis (GRH).

Prior to this work, the asymptotic behavior of these moments was conjectured to be of the order $T(\log T)^{[K:\mathbb{Q}]k^2}$ for the unshifted case ($a_j=1, b_j=0$). While Soundararajan established almost sharp bounds for the Riemann zeta function ($K=\mathbb{Q}$) and Harper refined this to the conjectured order, the extension to Dedekind zeta functions faced limitations. Milinovich and Turnage-Butterbaugh proved a bound with an extra $\epsilon$ in the exponent of the logarithm for general Galois extensions. Hagen recently removed this $\epsilon$ but only for finite solvable Galois extensions. The gap remained for arbitrary finite Galois extensions, where the sharp bound was unknown.

Methodology
The authors adapt and refine the method introduced by Harper for the Riemann zeta function, which relies on approximating the logarithm of the zeta function by short Dirichlet polynomials and utilizing a "long-short" decomposition of the integration interval. The core technical innovation in this paper lies in the adaptation of this framework to the arithmetic structure of Dedekind zeta functions via the Chebotarev Density Theorem.

Key methodological steps include:

1. Approximation: Using GRH, the authors approximate $\log |\zeta_K(1/2 + it)|$ by a sum over prime ideals, specifically focusing on the contribution of rational primes that split completely in $K$.
2. Decomposition: The interval $[T, 2T]$ is partitioned into a "good" set $\mathcal{T}$ where the Dirichlet polynomials are small, and a union of "bad" sets $S(j)$ where they exceed certain thresholds.
3. Chebotarev Application: A crucial step involves evaluating sums of the form $\sum_{p \le x, p \text{ splits}} \frac{\cos(\alpha \log p)}{p}$. By applying the Chebotarev Density Theorem to the identity conjugacy class, the authors show that the density of primes splitting completely in $K$ is $1/[K:\mathbb{Q}]$ of the total density of primes.
4. Exponent Improvement: Direct application of Harper's method to $\zeta_K(s)$ without exploiting the Galois structure would yield a bound with the exponent $[K:\mathbb{Q}]^2 k^2$. By utilizing the splitting behavior of primes, the authors reduce the effective contribution of the prime sums, improving the exponent from $[K:\mathbb{Q}]^2 k^2$ to the conjecturally optimal $[K:\mathbb{Q}] k^2$.

Key Contributions and Results

• Theorem 1.1 (Main Result): Assuming GRH, the authors establish an upper bound for shifted moments of $\zeta_K(s)$ for any finite Galois extension $K/\mathbb{Q}$. The bound is:
  $$\int_0^T \prod_{j=1}^{2k} \left| \zeta_K\left(\frac{1}{2} + i(t + b_j)\right) \right|^{a_j} dt \ll T (\log T)^{[K:\mathbb{Q}](\sum a_j^2)/4} \prod_{1 \le i < j \le 2k} g(|b_i - b_j|)^{[K:\mathbb{Q}] a_i a_j / 2}$$
  where $g(x)$ is a correlation function defined piecewise based on the separation $|b_i - b_j|$. This result holds for arbitrary finite Galois extensions, not just solvable ones.

• Corollary 1.2: By setting $a_j=1$ and $b_j=0$, the authors recover the sharp upper bound for the unshifted moments:
  $$\int_0^T \left| \zeta_K\left(\frac{1}{2} + it\right) \right|^{2k} dt \ll_{K,k} T (\log T)^{[K:\mathbb{Q}]k^2}$$
  This extends Hagen's result from solvable extensions to all finite Galois extensions.

• Corollary 1.3 (Large Deviations): The paper derives an upper bound for the measure of the set where $\log |\zeta_K(1/2 + it)|$ exceeds a threshold $V \asymp \sqrt{\log \log T}$. Assuming GRH, the measure decays as $e^{-V^2/2}$, matching the Gaussian heuristic predicted by random matrix theory.

• Corollary 1.4 (Short Intervals): The authors apply their moment bounds to study the variance of the coefficients $r_K(n)$ (the number of ideals of norm $n$) in short intervals. They provide a conditional estimate for the variance of the error term in the mean-value estimate of $r_K(n)$, improving upon previous bounds for Galois extensions.

Significance
The paper claims to fill a significant gap in the analytic theory of $L$-functions by establishing the conjecturally optimal upper bound for moments of Dedekind zeta functions for the broad class of finite Galois extensions. The significance is twofold:

1. Generalization: It removes the solvability constraint required by Hagen's recent work, demonstrating that the sharp bound holds for any finite Galois extension.
2. Methodological Refinement: The authors explicitly state that the key to their improvement over a naive application of Harper's method is the use of the Chebotarev Density Theorem to account for the splitting behavior of primes. This allows them to reduce the exponent of the logarithmic factor from $[K:\mathbb{Q}]^2 k^2$ to $[K:\mathbb{Q}] k^2$.

The authors note that while their method relies heavily on the Galois structure, extending these sharp bounds to arbitrary (non-Galois) number fields remains an open problem requiring new ideas.