AI 中文总结
研究素数数字分布，结合素数定理等方法证明素数中数字出现概率渐近为\(\frac{1}{10}\)，建立内部数字定量等分布，阐明与其他相关问题区别，得出素数数字汇总后渐近等分布的结论。
AI 中文摘要
设\(S_n = \{p\in\mathbb{P}:p < 10^n\}\)，\(N_n\)表示\(S_n\)中素数的十进制数字总数，\(C_n(d)\)是数字\(d\in\{0,\ldots,9\}\)在这些数字中的出现次数，\(P_n(d)\)是数字\(d\)出现的概率。我们证明\(P_n(d)=\frac{C_n(d)}{N_n}=\frac{1}{10}+O(\frac{\log n}{n})\)，\(n\to\infty\)，对每个十进制数字\(d\)一致成立。论证完全无条件，结合了素数定理、埃尔德什 - 图兰偏差不等式以及关于素数指数和的经典沃恩 - 维诺格拉多夫估计。主要步骤建立了内部数字位置的定量等分布，而在对所有数字位置和素数长度进行平均后，十进制展开末尾附近对数级数量的特殊位置的影响渐近可忽略。因此，当汇总所有位置和小于\(10^n\)的所有素数中的十进制数字时，它们渐近等分布。我们还通过区分这种平均等分布结果与关于素数序列中逐点数字等分布、正态性和高阶数字相关性的更强且目前未解决的问题，阐明了定理的精确范围。
英文摘要
Let $S_n=\{p\in\mathbb{P}:p<10^n\}$, $N_n$ denote the total number of decimal digits occurring in the primes of $S_n$, $C_n(d)$ be the number of occurrences of a digit $d\in\{0,\ldots,9\}$ among those digits, and $P_n(d)$ be the probability of occurrence of a digit, $d$ among those digits. We prove that \[ P_n(d)=\frac{C_n(d)}{N_n} =\frac{1}{10} +O\!\left(\frac{\log n}{n}\right), \qquad n\to\infty, \] uniformly for every decimal digit $d$. The argument is entirely unconditional and combines the Prime Number Theorem, the Erdős--Turán discrepancy inequality, and classical Vaughan--Vinogradov estimates for exponential sums over primes. The principal step establishes quantitative equidistribution for interior digit positions, while the logarithmically many exceptional positions near the ends of the decimal expansion are shown to have asymptotically negligible influence after averaging over all digit positions and prime lengths. Consequently, the decimal digits occurring in primes, when pooled over all positions and all primes below $10^n$, become asymptotically equidistributed. We also clarify the precise scope of the theorem by distinguishing this averaged equidistribution result from the substantially stronger and presently unresolved questions concerning pointwise digit equidistribution, normality, and higher-order digit correlations in the sequence of prime numbers.
