一个求和式中的斐波那契、狄利克雷和高斯
Fibonacci, Dirichlet, and Gauss in a single sum
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中文总结 AI 辅助
研究分数部分和\(\sum_{k = 1}^{n}\{F_n/F_k\}\)，其渐近行为因\(n\)奇偶性而异，奇数\(n\)对应高斯圆问题，偶数\(n\)对应狄利克雷除数问题，还证明了二阶递推数列类似公式。
中文摘要 AI 辅助
我们研究分数部分和\(\sum_{k = 1}^{n}\{F_n/F_k\}\)，其中\(F_n\)是第\(n\)个斐波那契数。其渐近行为取决于\(n\)的奇偶性。对于奇数\(n\)，余数用高斯圆误差项表示；对于偶数\(n\)，用狄利克雷除数误差项表示。确定奇数斐波那契和的最优余数指数等同于高斯圆问题，偶数和的相应问题等同于狄利克雷除数问题。我们还证明了包括卢卡斯序列在内的二阶递推数列的类似公式，其中两种奇偶性的作用互换。
英文摘要
We study the fractional-part sums $\sum_{k=1}^{n}\{F_n/F_k\}$, where $F_n$ is the $n$th Fibonacci number. Their asymptotic behavior depends on the parity of $n$. For odd $n$, the remainder is expressed in terms of the Gauss circle error term. For even $n$, it is expressed in terms of the Dirichlet divisor error term. Thus determining the optimal remainder exponent for the odd Fibonacci sums is equivalent to the Gauss circle problem, while the corresponding question for the even sums is equivalent to the Dirichlet divisor problem. We also prove analogous formulas for a family of second-order recurrences, including the Lucas sequence, for which the roles of the two parities are exchanged.
