通过掷骰子命中k个素数
Hitting k primes by dice rolls
AI总结：
本文研究独立公平骰子无限掷骰序列中，累积和达到素数$k$次所需掷骰子次数的随机变量$L_k$，证明大$k$时其期望值为$(1+o(1))k\log_e k$，并给出$k \leq 100$时的分布计算结果。
摘要：
设$S=(d_1,d_2,d_3,\ldots)$是独立公平骰子的无限掷骰序列。对于整数$k \geq 1$，令$L_k=L_k(S)$为最小的$i$，使得存在$k$个整数$j \leq i$满足$\sum_{t=1}^j d_t$是素数。因此，$L_k$是表示累积和达到素数$k$次所需掷骰子次数的随机变量。已知$L_1$的期望值接近2.43。本文证明，当$k$很大时，$L_k$的期望值为$(1+o(1))k\log_e k$（其中$o(1)$项随$k$趋于无穷而趋近于0），还包含$k \leq 100$时$L_k$分布的计算结果。
英文摘要：
Let $S=(d_1,d_2,d_3, \ldots )$ be an infinite sequence of rolls of independent fair dice. For an integer $k \geq 1$, let $L_k=L_k(S)$ be the smallest $i$ so that there are $k$ integers $j \leq i$ for which $\sum_{t=1}^j d_t$ is a prime. Therefore, $L_k$ is the random variable whose value is the number of dice rolls required until the accumulated sum equals a prime $k$ times. It is known that the expected value of $L_1$ is close to $2.43$. Here we show that for large $k$, the expected value of $L_k$ is $(1+o(1)) k\log_e k$, where the $o(1)$-term tends to zero as $k$ tends to infinity. We also include some computational results about the distribution of $L_k$ for $k \leq 100$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
