Joint distribution of primes in multiple short intervals

Technical Summary: Joint Distribution of Primes in Multiple Short Intervals

Problem Statement
The paper investigates the statistical behavior of the weighted count of primes, denoted by $\psi(x)$, within multiple short intervals. While the Prime Number Theorem establishes the mean density of primes, the fluctuations of $\psi(x+H) - \psi(x)$ in short intervals have been a subject of intense study. Previous work by Montgomery and Soundararajan established that for intervals of length $H$ where $H \to \infty$ and $\log H / \log N \to 0$, the distribution is asymptotically normal. However, the joint distribution of primes in multiple disjoint short intervals, and specifically the correlations between them, remained less explored, particularly in the context of "multiplicative" intervals where the length scales with the position $x$ (i.e., $h = \delta x$).

The central questions addressed are:

1. What is the joint limiting distribution of the weighted prime counts in $r$ disjoint short intervals as the upper bound $X \to \infty$?
2. How are these counts correlated?
3. Do these correlations induce biases in "prime number races" (comparisons of prime counts between intervals) similar to the Shanks–Rényi prime number race in arithmetic progressions?

Methodology
The author adopts a Fourier-analytic approach, adapting the method of Rubinstein and Sarnak (originally developed for primes in arithmetic progressions) to the setting of short intervals. The core assumptions are the Riemann Hypothesis (RH) and the Linear Independence (LI) conjecture regarding the positive ordinates of the non-trivial zeros of the Riemann zeta function.

Key methodological steps include:

• Explicit Formula: Utilizing the explicit formula to relate the prime counting function $\psi(x)$ to the zeros $\rho = 1/2 + i\gamma$ of the zeta function.
• Random Model Construction: Under RH and LI, the normalized deviation of the prime count is modeled as a random vector $X_{\delta, t}$ involving a sum over the zeros, weighted by a specific function $w(\rho)$ and multiplied by independent random variables $U_\gamma$ uniformly distributed on the unit circle.
• Logarithmic Density: The distribution is analyzed with respect to logarithmic density ($\frac{1}{\log X} \int_2^X \dots \frac{dx}{x}$), which is natural for multiplicative intervals.
• Characteristic Functions: The proof relies heavily on analyzing the Fourier transform (characteristic function) of the limiting measure. The author derives an infinite product of Bessel functions ($J_0$) representing this transform.
• Asymptotic Analysis: By estimating the covariance matrix and controlling the error terms in the Fourier inversion formula, the author demonstrates convergence to a multivariate Gaussian distribution. The analysis involves delicate estimates of sums over zeros, utilizing the Riemann–von Mangoldt formula and properties of the function $\Delta(t)$, which describes the repulsion between intervals.

Key Contributions and Results

1. Multivariate Gaussian Distribution:
   The paper proves that under RH and LI, the weighted count of primes in $r$ disjoint short intervals of length $\delta x$ (where $\delta$ is small but fixed relative to $X$) follows a multivariate Gaussian distribution. The mean is zero (after normalization), and the covariance structure is explicitly determined.

2. Weak Negative Correlations:
   A primary finding is that the counts in disjoint short intervals are not independent; they exhibit weak negative correlations. The covariance between two intervals separated by a parameter distance $|t_j - t_k|$ is approximately proportional to $-\frac{\Delta(|t_j - t_k|)}{\log(1/\delta)}$. The function $\Delta(t)$ behaves like $1/(2t)$ for large $t$. This implies a "Coulomb-like" repulsion: primes in one interval slightly suppress the likelihood of primes in a neighboring interval.

3. Phase Transition in Prime Number Races:
   Applying these distributional results to "prime number races" (ordering the counts of primes in multiple intervals), the paper identifies a sharp phase transition:

   • Unbiased Regime: If the number of intervals $r$ is small relative to $\log(1/\delta)$ (specifically $r = o(\frac{\log(1/\delta)}{\log \log(1/\delta)})$), the races remain asymptotically unbiased; all orderings occur with roughly equal probability ($1/r!$).
   • Biased Regime: Once $r$ exceeds a critical threshold (roughly proportional to $\frac{\log(1/\delta)}{\log \log(1/\delta)}$), biases emerge. The paper shows that for sufficiently large $r$, there exist orderings of the intervals that occur with significantly lower probability than others. This is a direct consequence of the accumulated negative correlations.

4. Quantitative Linear Independence (QLI) Extension:
   For a single moving interval where the length parameter $\delta$ tends to zero as $X \to \infty$ (specifically $\delta > (\log X)^{-\varepsilon}$), the author proves a central limit theorem assuming a quantitative formulation of the LI conjecture (QLI). This extends the validity of the Gaussian distribution to shorter intervals than previously established under qualitative LI alone.

Significance and Claims
The paper claims to be the first to focus on the distribution of primes in short intervals using the Rubinstein–Sarnak framework, shifting the focus from residue classes to interval lengths.

• Novelty: The result is new even for a single moving interval under a quantitative LI formulation.
• Mechanism of Bias: The work provides a transparent explanation for the emergence of bias in prime number races in short intervals, attributing it to the specific structure of the covariance matrix (weak negative correlations) and the aggregation of these correlations as the number of intervals grows.
• Sharpness: The author notes that the derived critical threshold for the phase transition is sharper than previous results in the context of the Shanks–Rényi race, owing to the precise correlation structure and refined estimates for the determinant of the covariance matrix.
• Limitations: The paper acknowledges that the Berry–Esseen type arguments used impose a limit on the number of intervals $r$ (preventing it from reaching a full power of $\delta^{-1}$). The author suggests that alternative methods (Stein's method or Lindeberg type) might extend this range, but the current Fourier analytic approach was chosen to establish tight bounds on the total variation distance.

In summary, the paper establishes a rigorous probabilistic framework for primes in multiple short intervals, revealing a subtle repulsive force between intervals that leads to a phase transition from unbiased to biased behavior in comparative prime counting.