Bourgade's Central Limit Theorem

• Bourgade's Central Limit Theorem is an extension of Selberg’s classical result, establishing multivariate Gaussian limit laws for log-shifted Dirichlet L-functions.
• It employs analytic techniques such as truncated Euler products and zero-density estimates to derive quantitative convergence rates in the Dudley metric.
• The theorem reveals that arithmetic correlations and logarithmic shifts critically shape the covariance structure, influencing the convergence speed of the Gaussian approximation.

Bourgade’s Central Limit Theorem refers to an advanced multivariate central limit theorem (CLT) for log values of shifted Dirichlet $L$-functions, extending the classical Selberg CLT and revealing intricate dependence structures in random vectors built from analytic number theory objects. The statement and quantitative rates of convergence for Bourgade’s CLT in Dirichlet $L$-functions are elucidated by Hsu–Wong (Hsu et al., 6 Jan 2026), with refined analysis based on works of Radziwiłł–Soundararajan and Roberts. The theorem establishes Gaussian limit laws for vectors assembled from $\log|L(½+i(U+\alpha_j),\chi_j)|$ for independent Dirichlet characters $\chi_j$ and real shifts $\alpha_j$, demonstrating how component correlations critically affect convergence rates.

1. Model Setup and Formulation

Let $N\geq1$ be fixed. Consider primitive Dirichlet characters $\chi_1,\dots,\chi_N$ (respective moduli $q_j$), and shifts $\alpha_1(T),\dots,\alpha_N(T)$ satisfying $|\alpha_j(T)|\leq T/2$. For a random $U=U_T$ uniformly distributed in $[T,2T]$, define the $N$-vector random variable

$$X_T(U) = \left( X_{j,T}(U) \right)_{1\leq j \leq N}, \quad X_{j,T}(U) = \frac{\log | L(½+i(U+\alpha_j), \chi_j) |}{\sqrt{\log\log T}}.$$

Bourgade’s theorem (originally established in the zeta-function setting) asserts that under spacing conditions on the shifts $\alpha_j$, the vector $X_T(U)$ converges in distribution as $T\to\infty$ to a centered $N$-variate Gaussian $\widetilde{X}\sim N(0,K)$, where $K$ is the covariance matrix determined by the log-distance of shifts and character-twist relations:

• For $1\leq i<j\leq N$, $\delta_{i,j}=1$ if $\chi_i\overline{\chi_j}$ is principal (else $0$).
• $c_{i,j}\in[0,1]$ is defined by matching

$$\log\left(\frac{1}{|\alpha_i-\alpha_j|}\right) = c_{i,j}\log\log T + O((\log\log\log T)^\varepsilon).$$

• Covariance matrix entries: $k_{i,i}=1$, $k_{i,j}=\delta_{i,j}c_{i,j}$ for $i\neq j$.

This formalizes the multivariate CLT for shifted Dirichlet $L$-functions, conditional on $K$ being positive-definite.

2. Hypotheses and Dependence Structure

The hypothesis requires each $|\alpha_j| \leq T/2$, and pairwise spacings $$|\alpha_i-\alpha_j| \geq \exp(-O((\log\log T)^\varepsilon))$$. The matrix $K$ captures both the arithmetic (via $\delta_{i,j}$, "principal twist") and the geometric (via $c_{i,j}$, log-distance) dependence. Hence, only "like" characters (those sharing a primitive part) correlate; $k_{i,j}$ measures the strength accordingly. With distinct quadratic characters, partial correlations $c_{i,j}/2$ arise. The dependence encoded in $K$ directly governs the joint limiting distribution and rate of convergence.

3. Rates of Convergence in Dudley Metric

Rates are quantified via the Dudley (bounded-Lipschitz) metric $d_D(\cdot,\cdot)$ on $N$-vectors, parameterized by bounds $L,M$ on Lipschitz constants and sup-norms:

• General dependent case (Theorem 1.3):

For any $0<\varepsilon_1<\varepsilon_2$ with $\varepsilon_1+\varepsilon_2<1$, for sufficiently large $T$,

$$d_D( X_T, \widetilde{X} ) \ll_{K,N} \frac{L}{(\log\log\log T)^{\varepsilon_1}} + M (\log\log\log T)^{N(\varepsilon_1+\varepsilon_2)} \exp\left( -\tfrac12(\log\log\log T)^{\varepsilon_1+\varepsilon_2} \right)$$

so $d_D \to 0$ as $T\to\infty$.

• Independent case $(\Delta=0, \delta=0)$ for $N=1,2,3$ (Theorem 1.4):
  • For $N=1,2$:

  $$d_D(X_T, \widetilde{X}) \ll \frac{L\,N(\log\log\log T)^2}{ \sqrt{\log\log T} } + \frac{M}{(\log\log T)^{1-\varepsilon-\varepsilon_3}}$$ - For $N=3$:

  $$d_D(X_T, \widetilde{X}) \ll \frac{L\,N(\log\log\log T)^2}{ \sqrt{\log\log T} } + \frac{M}{(\log\log T)^{\tfrac12-\varepsilon_4}}$$

These recover and extend Selberg’s $O((\log\log T)^{-1/2})$ rate in the univariate case to independent multivariate settings.

4. Proof Architecture and Approximation Steps

The convergence analysis employs a seven-step approximation scheme:

1. Express $X_T$ via $\log|L(½+i(U+\alpha_j))|/\sqrt{\log\log T}$.
2. Shift $s=½+i(U+\alpha_j)$ to $\sigma_0+i(U+\alpha_j)$ with $\sigma_0=½+W/\log T$, $W\sim (\log\log\log T)^2$; Dudley error $O(LW/\sqrt{\log\log T})$.
3. Truncate Euler product to Dirichlet polynomial $M(s)$ of controlled length $T^{o(1)}$, using zero-density estimates (Radziwiłł–Soundararajan) to manage error $O(M(\Delta+\delta)/\sqrt{\log\log T})$.
4. Approximate $\log M^{-1}$ by $P(s)=\sum_{p\leq X}\chi(p)p^{-s}$ via Mertens/log expansions.
5. Renormalize $P(s)$ to match the covariance structure, yielding $R^1_T$.
6. Apply cumulant/moment bounds (Roberts’ method) to compare characteristic functions of $R^1_T(U)$ and the target Gaussian, yielding the core exponential error.
7. Use matrix perturbation arguments (Taylor expansion and determinant comparison) to compare $\widetilde{X}$ with the Gaussian.

Steps 3–4 exploit moments methods and avoid zeros of $L$; step 6 generalizes Roberts’ Stein/CF technique to multivariate dependencies, with dimensional tracking.

5. Influence of Dependence on Rates

The rate in Theorem 1.3 is governed by the number of components $N$ and the dependence structure, as encoded by $k_{i,j}=\delta_{i,j}c_{i,j}$. The measure decays only doubly logarithmically, then exponentially in $(\log\log\log T)^{\varepsilon}$, with the pre-factor $(\log\log\log T)^{N(\varepsilon_1+\varepsilon_2)}$ arising from the multivariate setting and the degree of correlation. In contrast, independence ($\Delta = \delta = 0$) leads to much faster polylogarithmic rates, matching Selberg’s original results in the univariate case and extending them for $N=2,3$. This demonstrates that even mild logarithmic correlations among components can drastically slow multivariate convergence.

6. Context, Extensions, and Related Results

The Bourgade CLT generalizes the classical Selberg result to multivariate, correlated settings and Dirichlet $L$-functions. The Hsu–Wong results elaborate this theorem with precise metric rates, fully quantifying the impact of dependence and the number of components. The reliance on recent advances by Radziwiłł–Soundararajan permits effective control of the error via zero-density estimates, while Roberts’ techniques enable detailed moment/cumulant bounds in dependent regimes. A plausible implication is that the structure of arithmetic and geometric correlations in high-dimensional vector-valued analytic number theory profoundly influences the practical speed of Gaussian approximation in central limit phenomena (Hsu et al., 6 Jan 2026).