短区间内素数分布的偏差
Biases in the distribution of primes in short intervals
AI总结：
在Hardy-Littlewood素数元组假设下，研究短区间内素数计数分布的二级渐近，发现算术修正导致比Cramér模型更强的向均值偏差，并通过有限筛论证和数值比较验证。
AI中文摘要：
在适当均匀的Hardy-Littlewood素数元组假设下，当区间长度与平均素数间距相当时，我们获得了包含指定数量素数的短区间比例的二级渐近式。首项为泊松分布，但算术修正不同于Cramér独立模型中的二项修正，并预测了向均值附近计数更强的偏差。证明结合了容斥原理与Montgomery和Soundararajan的奇异级数估计，以及一个有限筛论证，该论证在截断阶增长时控制所需的交错和。我们还给出了一个精化的随机模型来重现该修正，以及支持该预测的数值比较。
英文摘要：
Assuming a suitably uniform Hardy--Littlewood prime tuples hypothesis, we obtain a second-order asymptotic for the proportion of short intervals containing a prescribed number of primes, when the interval length is comparable to the average prime spacing. The leading term is Poisson, but the arithmetic correction differs from the binomial correction in Cramér's independent model and predicts a stronger bias toward counts near the mean. The proof combines inclusion--exclusion with singular-series estimates of Montgomery and Soundararajan and a finite-sieve argument that controls the required alternating sums as the truncation order grows. We also give a refined random model that reproduces the correction and numerical comparisons that support the prediction.
