On the $\beta=2$ Partition function for Dirichlet $L$-functions in the $q$-aspect

Technical Summary: On the $\beta = 2$ Partition Function for Dirichlet $L$-Functions in the $q$-Aspect

Problem Statement
This paper investigates the size of Dirichlet $L$-functions, $L(s, \chi)$, for typical Dirichlet characters $\chi$ modulo a large prime $q$, specifically near the central point $s=1/2$. The primary object of study is the $\beta = 2$ partition function over mesoscopic intervals:
$$Z_2(q, \chi, \theta) = \int_{|h| \le \log^\theta(q)/2} |L(1/2 + ih, \chi)|^2 dh$$
where $\theta \in (-1/2, 0]$. The work is motivated by $q$-analogues of the Saksman–Webb and Fyodorov–Hiary–Keating (FHK) conjectures, which predict specific normalizations and limiting distributions for the maximum and moments of these $L$-functions. A central challenge addressed is determining the correct normalization factor (involving $\sqrt{\log \log q}$) and establishing sharp upper bounds for the typical maximum of $|L(1/2+ih, \chi)|$.

Methodology
The proofs rely on a combination of analytic number theory and probabilistic methods, adapting techniques developed by Harper, Arguin, and others for the Riemann zeta function to the $q$-aspect.

1. Randomization Argument: The core technique involves transferring deterministic character sums to sums involving Steinhaus random multiplicative functions $f(n)$. Using orthogonality of Dirichlet characters and specific moment estimates (Lemma 2 and Lemma 3), the author approximates averages over characters with expectations over random multiplicative functions. This allows the use of probabilistic tools that are otherwise difficult to apply directly to deterministic $L$-functions.
2. Approximate Functional Equation and Factorization: The $L$-function is approximated using the approximate functional equation, splitting the resulting Dirichlet polynomial into components based on the prime factorization of the integers involved:
   • Smooth part: Integers with all prime factors $\le P_\theta = \exp(\log^{|\theta|} q)$.
   • Rough part: Integers with all prime factors $> P_\theta$.
   • Intermediate part: Integers with prime factors in a specific range.
     This factorization separates the "log-normal" contribution (from smooth numbers) from the "log-correlated" contribution (from rough numbers).
3. Explicit Conditioning: A novel aspect of this work is the handling of mesoscopic intervals ($\theta < 0$). The author introduces explicit conditioning on the value of the Dirichlet polynomial over the smooth numbers, denoted $Y_{P_\theta}(0)$. This conditioning is necessary because the smooth part is essentially "frozen" over the interval, and its value dictates the magnitude of the integral. The paper reformulates implicit conditioning used in previous works (e.g., Harper [Har23]) into an explicit one-point conditioning statement using a chaining argument.
4. Chaining and Barrier Events: To control the maximum of the log-correlated part, the paper employs a chaining argument (Proposition 3.1) and defines barrier events (e.g., $G_\chi$) that restrict the growth of partial Euler products. These events mimic the behavior of Gaussian multiplicative chaos.
5. Cauchy Integral Formula: For the proof of the maximum bound (Theorem 3), the author approximates the value of $L$ at a typical maximum using a Cauchy integral over a small rectangle. This reduces the problem to estimating fourth moments or $\beta=2$ partition functions over specific ranges.

Key Contributions and Results

• Theorem 1 (Moments of the Partition Function): For $\theta = 0$, the paper establishes moment bounds for the partition function $Z_2(q, \chi, 0)$. It proves that for $r \in [0, 1]$, the average of the $r$-th moment is bounded by:
  $$\frac{1}{q-1} \sum_{\chi \pmod q} (Z_2(q, \chi, 0))^r \ll \left( \frac{\log q}{1 + (1-r)\sqrt{\log \log q}} \right)^r$$
  This provides evidence that the normalization factor $\sqrt{\log \log q}$ in the Saksman–Webb conjecture is correct.

• Theorem 2 (Mesoscopic Partition Function with Conditioning): For $\theta \in (-1/2, 0]$, the paper derives moment bounds for $Z_2(q, \chi, \theta)$ conditioned on the value of the smooth part $Y_{P_\theta}(0)$. The bounds depend explicitly on the conditioning parameter $W$ and the interval size $\theta$:
  $$\frac{1}{q-1} \sum_{\chi \pmod q} \left[ (Z_2(q, \chi, \theta))^r \mid Y_{P_\theta}(0) \in [W, W+1] \right] \ll \left( \frac{e^{2W} \log^{1+2\theta} q}{1 + (1-r)\sqrt{\log \log q}} \right)^r$$
  This result generalizes previous bounds for the Riemann zeta function to the $q$-aspect and handles the explicit conditioning required for shrinking intervals.

• Corollary 1 (Tail Bounds): The moment bounds imply tail bounds for the partition function, confirming the order of magnitude predicted by Conjecture 1.

• Theorem 3 (Upper Bound for Typical Maxima): The paper proves an upper bound for the maximum of $|L(1/2+ih, \chi)|$ over $|h| \le 1/2$ for typical characters. Specifically, for $0 \le U \le \log \log q$:
  $$\frac{1}{q-1} \# \left\{ \chi \pmod q : \max_{|h| \le 1/2} |L(1/2+ih, \chi)| \ge \frac{e^U \log q}{(\log \log q)^{3/4}} \right\} \ll e^{-2U} (\log \log \log q)^2 (\log \log \log q + U)$$
  This establishes an upper bound that matches the predictions of the $q$-analogue of the Fyodorov–Hiary–Keating conjectures up to the second-order correction term, up to additional logarithmic factors.

Significance and Claims
The paper claims to provide an upper bound for the typical maxima of Dirichlet $L$-functions in the $q$-aspect that matches the FHK conjecture predictions up to the second-order correction term (with additional logarithmic factors). It demonstrates that the $\beta=2$ partition function is a crucial tool for understanding these maxima, as it allows access to "ballot theorem" savings that are unavailable at smaller $\beta$.

The author notes that while the upper bound is sharp up to the second-order term, the paper does not pursue matching lower bounds; proving the full conjecture (including lower bounds) would require more delicate handling of Dirichlet polynomials over rough numbers or a full proof of Conjecture 1, which involves significant algebraic challenges (such as computing twisted fourth moments). The work also highlights the necessity of explicit conditioning on smooth numbers when dealing with mesoscopic intervals, a technique that differs from previous approaches that relied on level sets of $\log |L|$.

The paper concludes that the normalization factor $\sqrt{\log \log q}$ is indeed the correct order of magnitude for the critical partition function, a result consistent with the theory of critical Gaussian multiplicative chaos.