Lower bounds for the large deviations and moments of the Riemann zeta function on the critical line

1. Problem Statement and Context

The paper addresses the distribution of values of the Riemann zeta function, $\zeta(s)$, on the critical line $\text{Re}(s) = 1/2$. Specifically, it investigates large deviations of $\log|\zeta(1/2 + it)|$ and the fractional moments of the zeta function.

• Background: Selberg's Central Limit Theorem establishes that $\log|\zeta(1/2 + it)|$ behaves like a Gaussian random variable with mean 0 and variance $\log \log T$ for $t \in [0, T]$.
• The Gap: While the Gaussian behavior is well-understood for deviations of order $\sqrt{\log \log T}$, the behavior for larger deviations (large deviations) and the precise constants governing the growth of moments $M_{2\alpha}(T) = \int_0^T |\zeta(1/2 + it)|^{2\alpha} dt$ have been less settled unconditionally.
• Previous Work:
  • Soundararajan [Sou09] provided lower bounds for large deviations and moments but relied on the Riemann Hypothesis (RH) for the upper bounds and specific constants.
  • Heuristic models (Keating-Snaith) and conditional results suggest specific asymptotic constants involving arithmetic and geometric factors.
  • Recent work by the authors [AB23, AB24] established upper bounds and lower bounds off the critical line, but the critical line case with optimal constants remained open.

The Goal: To prove an unconditional lower bound for the measure of level sets where $\log|\zeta(1/2 + it)| > V$ (with $V \sim \alpha \log \log T$) that matches the order of magnitude of known upper bounds and recovers the sharpest known constants for fractional moments without assuming RH.

2. Methodology

The proof employs a sophisticated recursive scheme combining probabilistic methods, Dirichlet polynomial approximations, and moment estimates. The core strategy involves the following steps:

A. Approximation by Partial Sums

The authors approximate $\log|\zeta(1/2 + it)|$ using real parts of Dirichlet polynomials (partial sums of primes). They define partial sums $S(\eta, t)$ over primes up to a cutoff $T^* = T^{1/R(\alpha^2+1)}$.

• Truncation: Unlike previous works that truncated at much smaller scales, this method pushes the truncation point $T^*$ as high as possible while maintaining control over the error terms.
• Decomposition: The interval of primes is divided into dyadic blocks $P_j$ on a log-log scale. The partial sums $S_j(t)$ over these blocks are treated as steps in a random walk.

B. The "Good Event" and Barriers

To establish a lower bound, the authors define a "good event" $G_J$ where the partial sums $S_j(t)$ stay within a narrow "barrier" (a tube) around a linear trajectory with a specific gradient $\kappa^*$.

• Barrier Definition: The barriers $[L_j(z), U_j(z)]$ are constructed to ensure that if the walk stays within them, the final value $\log|\zeta|$ exceeds the target $V$.
• Mollification: To handle the rare event $G_J$, the authors use a mollifier $M(t)$ (a Dirichlet polynomial) designed to dampen the large values of $\zeta$ and restrict the analysis to the "good" region.

C. The Paley-Zygmund Inequality

The core probabilistic tool is the Paley-Zygmund inequality, which provides a lower bound for the probability that a non-negative random variable $Z$ exceeds a fraction of its mean:
$$P(Z \ge \delta E[Z]) \ge (1-\delta)^2 \frac{E[Z]^2}{E[Z^2]}$$
Here, $Z = |\zeta M|^2$ restricted to the event $G_J$. To make this effective, the authors must estimate the conditional second and fourth moments of the mollified zeta function.

D. Twisted Moments and Tilting

A critical technical innovation is the use of polynomial tilts rather than simple indicator functions.

• Tilting: To make the rare event $G_J$ "typical" in a new measure, the authors introduce a tilt based on powers of the partial sums. This allows them to use shorter Dirichlet polynomials for the twist, which is essential for applying the Mean Value Theorem for Dirichlet polynomials.
• $\alpha$-Separable Polynomials: They define a class of polynomials ($\alpha$-separable) that allow the twisted moments to be decoupled across different prime intervals. This structure is crucial for proving that the twisted moments behave like those of a Gaussian random walk (the Steinhaus model).
• Approximation of Indicators: Instead of using sharp indicator functions (which require high-degree polynomials), they approximate the barriers using polynomials $D_\pm$ (based on Lemma 3.2) that closely mimic the indicator functions but have controlled growth outside the target interval.

3. Key Contributions and Results

Theorem 1.1 (Main Result: Large Deviations)

The authors prove that for any $\alpha > 0$ and $V \sim \alpha \log \log T$, the measure of the set where $\log|\zeta(1/2 + it)| > V$ satisfies:
$$\frac{1}{T} \text{meas}\left\{ t \in [0, T] : \log|\zeta(1/2 + it)| > V \right\} \ge K_\alpha \int_V^\infty \frac{e^{-y^2/\log \log T}}{\sqrt{\log \log T}} dy$$
where the constant $K_\alpha$ satisfies:
$$K_\alpha \gg (C \alpha^2 \log \alpha)^{-\alpha^2}$$

• Significance: This is an unconditional lower bound. It matches the order of magnitude of the upper bound proved in [AB23] for $0 < \alpha < 2$.
• Constant Improvement: The constant $K_\alpha$ is sharper than previous unconditional results (which had a worse dependence like $\alpha^{-36}$). It approaches the conjectured value derived from the Keating-Snaith moments conjecture, differing only by a factor related to the arithmetic factor $a_\alpha$.

Corollary 1.2 (Fractional Moments)

By integrating the large deviation bound, the authors recover the lower bound for the fractional moments of the zeta function:
$$\int_0^T |\zeta(1/2 + it)|^{2\alpha} dt \gg K_\alpha T (\log T)^{\alpha^2}$$
This confirms Soundararajan's heuristic that the dominant contribution to the $2k$-th moment comes from values of size $(\log T)^k$ occurring with measure $T/(\log T)^{k^2}$.

Theorem 1.3 (The $q$-aspect)

The method is generalized to Dirichlet $L$-functions. They prove a similar lower bound for the central values $L(1/2, \chi)$ of primitive even characters modulo a large prime $q$:
$$\frac{1}{\phi^+(q)} \text{meas}\left\{ \chi : \log|L(1/2, \chi)| \ge W \right\} \ge K_\alpha \int_W^\infty \dots$$
This yields lower bounds for the moments of central $L$-values, consistent with the Central Limit Theorem for $L$-functions.

4. Technical Significance and Novelty

1. Unconditional Sharpness: The paper removes the dependency on the Riemann Hypothesis for these specific lower bounds, providing the sharpest known unconditional constants.
2. Optimal Truncation: By using a tilted probability measure combined with polynomial approximations, the authors can control the partial sums up to $T^* = T^{1/R(\alpha^2+1)}$. Previous methods relying on indicator functions required truncation at much smaller scales ($T^{1/\alpha^m}$), leading to worse constants.
3. Refined Moment Analysis: The proof of Proposition 3.5 (Twisted Mollifier Formulae) is a major technical achievement. It establishes precise asymptotics for the second and fourth twisted moments of $\zeta$ multiplied by $\alpha$-separable polynomials, effectively bridging the gap between the deterministic zeta function and the probabilistic Steinhaus model.
4. Handling the "Small Primes": The authors carefully distinguish between small primes ($P_0$) and large primes. For small primes, they retain the full Euler product structure to capture the arithmetic factor $a_\alpha$, while for large primes, they rely on Gaussian approximation.

5. Conclusion

Arguin and Creighton have successfully established that the large deviations of the Riemann zeta function on the critical line follow a Gaussian tail with the correct constant factor, unconditionally. This result not only validates the heuristic predictions regarding the distribution of zeta values but also provides a robust framework for analyzing moments of $L$-functions in the $q$-aspect. The work represents a significant step forward in analytic number theory by unifying probabilistic heuristics with rigorous, unconditional analytic bounds.