Conditional Upper Bounds for Large Deviations and Moments of the Riemann Zeta Function

1. Problem Statement

The paper addresses the distribution of large values of the Riemann zeta function, $\zeta(s)$, on the critical line $\text{Re}(s) = 1/2$. Specifically, it seeks to establish sharp upper bounds for:

1. Large Deviations: The measure of the set of $t \in [T, 2T]$ where $|\zeta(1/2 + it)|$ exceeds a threshold of order $(\log T)^k$ (or equivalently, $\log|\zeta(1/2+it)| > V$ where $V \sim k \log \log T$).
2. Moments: The $2k$-th moments of $|\zeta(1/2+it)|$, defined as $\frac{1}{T} \int_T^{2T} |\zeta(1/2+it)|^{2k} dt$.

Context:

• Selberg's Central Limit Theorem establishes that $\log|\zeta(1/2+it)|$ behaves like a Gaussian random variable with variance $\frac{1}{2}\log \log T$ for fixed $v$.
• Soundararajan (2009) proved that under the Riemann Hypothesis (RH), the measure of large deviations is roughly $(\log T)^{-k^2}$.
• Harper (2013) sharpened the bound on the moments to $\ll e^{O(k)} (\log T)^{k^2}$.
• The Gap: While unconditional bounds exist for small $k$ ($0 \le k \le 2$), extending Harper's sharp moment bounds to larger $k$ and establishing precise large deviation estimates for the full range of $V \sim k \log \log T$ under RH remained an open challenge. The optimal constant in these bounds is predicted by the Keating-Snaith conjecture to involve arithmetic and geometric constants ($a_k g_k$) that decay rapidly as $k \to \infty$.

2. Methodology

The authors employ a sophisticated probabilistic approach combining Soundararajan's conditional pointwise bound with a recursive scheme previously developed in [ABR20].

A. Soundararajan's Pointwise Bound

The proof relies on Proposition 1.3 (Soundararajan), which states that under RH, for a parameter $x$, $\log|\zeta(1/2+it)|$ is bounded by the real part of a Dirichlet polynomial plus a remainder term:
$$\log |\zeta(1/2+it)| \le \text{Re} \sum_{n \le x} \frac{a(n)\Lambda(n)}{n^{\sigma+it}\log n} + \text{Error}$$
The challenge is to choose $x$ large enough to capture the main contribution to the moments while maintaining control over the Dirichlet polynomial to prove Gaussian behavior.

B. The Recursive Scheme (Decomposition of Events)

The core innovation is the decomposition of the "bad" event $H = \{ \log|\zeta(1/2+i\tau)| > V \}$ into a sum of "good" events $G_\ell$.

1. Checkpoints: The interval $[T, 2T]$ is analyzed through a sequence of checkpoints $T_\ell = T^{\beta_\ell}$, where $\beta_\ell$ decreases exponentially.
2. Partial Sums: At each checkpoint, the authors define partial sums $S^{(j)}_\ell$ of the Dirichlet polynomial. These sums are modeled as random walks with time index $t_\ell = \log \log T_\ell$.
3. Good Events ($G_\ell$): The event $G_\ell$ restricts the partial sums to stay within a "tube" around a linear interpolation of the target value $V$. Specifically, $S^{(j)}_\ell \in [L_\ell, U_\ell]$, where the barriers are:
   $$L_\ell = \kappa \log \log T_\ell - c_\ell, \quad U_\ell = \kappa \log \log T_\ell + c_\ell$$
   Here, $\kappa = V / \log \log T$ is the gradient, and $c_\ell$ represents the allowed fluctuations (barrier width).
4. Barrier Width: A critical technical choice is the width $c_\ell = \beta_\ell^{-\gamma}$ (with $\gamma = 1/25$). This width narrows as $\ell$ increases, forcing the random walk to stay close to the linear trend required to reach the large deviation $V$.

C. Estimation Techniques

To bound the probability of the event $H \cap (G_{\ell-1} \setminus G_\ell)$ (i.e., the walk exits the tube at step $\ell$), the authors split the analysis into:

• Upper Barrier: Using union bounds and Gaussian tail estimates for the increments of the Dirichlet polynomials.
• Lower Barrier (The Novelty): This is the most difficult part. The authors use Soundararajan's bound (Proposition 1.3) to relate the tail of $\log|\zeta|$ to the partial sums. They condition on the position at step $\ell$ and show that if the sum drops below the lower barrier, the remaining tail of the zeta function cannot compensate to reach the total height $V$. This involves a careful analysis of the discrepancy between different abscissae ($\sigma_j$ vs $\sigma_m$) in the Dirichlet sums.
• Moments of Dirichlet Polynomials: The proofs utilize moment estimates (Appendix A) comparing Dirichlet polynomials to independent Steinhaus random variables (random phases) to approximate Gaussian behavior.

3. Key Contributions

1. Conditional Large Deviation Bound: The paper proves Theorem 1.1, establishing that under RH, for $V \sim k \log \log T$:
   $$\frac{1}{T} \text{meas} \{ t \in [T, 2T] : \log|\zeta(1/2+it)| > V \} \le C_k \frac{e^{-V^2 / \log \log T}}{\sqrt{\log \log T}}$$
   where $C_k = \exp(e^{ck})$. This confirms that Gaussian fluctuations extend to the large deviation regime for the entire range of $V \sim k \log \log T$.
2. Sharp Moment Bounds: By integrating the large deviation bound, the authors recover Harper's bound for the $2k$-th moments (Corollary 1.2):
   $$\frac{1}{T} \int_T^{2T} |\zeta(1/2+it)|^{2k} dt \ll e^{eck} (\log T)^{k^2}$$
   This holds for $k \ge 2$ (previously known for $k < 2$ unconditionally and $k \ge 2$ conditionally but with less precise constants).
3. Refined Recursive Scheme: The paper improves upon the [ABR20] scheme by handling the "lower barrier" problem more effectively using Soundararajan's bound, allowing for larger $k$ and tighter control over the Dirichlet polynomial truncation.
4. Short Interval Applications: The results are extended to short intervals of length $(\log T)^\gamma$, providing conditional upper bounds for the maximum of $|\zeta|$ and its moments in these intervals (Corollaries 3.1 and 3.2).

4. Main Results

• Theorem 1.1: Provides the conditional upper bound for the measure of large values. The decay rate matches the Gaussian prediction $e^{-V^2 / \log \log T}$, confirming the "Gaussian tail" behavior even for large deviations.
• Corollary 1.2: Establishes the moment bound $\ll e^{O(k)} (\log T)^{k^2}$. While the constant $C_k$ is not yet the optimal $a_k g_k$ predicted by Keating-Snaith (which decays to 0 as $k \to \infty$), the exponential dependence on $k$ is controlled, and the power of $\log T$ is exactly $k^2$.
• Corollary 3.1 & 3.2: Extend these bounds to short intervals, showing that the maximum of $|\zeta|$ on intervals of length $(\log T)^\gamma$ is bounded by $(\log T)^{\sqrt{1+\gamma}}$ (up to logarithmic factors), conditional on RH.

5. Significance

• Validation of Gaussian Heuristics: The work provides rigorous evidence that the Gaussian Central Limit Theorem for $\log|\zeta|$ is robust and extends to the large deviation regime, a non-trivial result given the arithmetic nature of the zeta function.
• Advancement of the Moments Problem: It pushes the boundary of known moment bounds under the Riemann Hypothesis. While the optimal constant remains elusive (requiring control of primes in the range $T^{1/k^2}$ to $T^{1/k}$), this paper narrows the gap significantly.
• Methodological Impact: The recursive scheme combined with Soundararajan's bound offers a powerful new toolkit for analyzing extreme values of $L$-functions. It suggests that similar techniques could be applied to other families of $L$-functions to derive sharp large deviation estimates.
• Short Intervals: The extension to short intervals is crucial for understanding the local behavior of the zeta function, which is relevant to the Fyodorov-Hiary-Keating conjecture regarding the maximum of the zeta function.

In summary, this paper represents a significant step forward in the analytic number theory of the Riemann zeta function, successfully bridging the gap between classical moment estimates and the conjectured behavior of large deviations under the assumption of the Riemann Hypothesis.