发表机构
Bishop’s University（主教大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究对费马大定理相关多项式的倒数有限和进行拆分分析，发现其近似线性且斜率与黎曼ζ函数相关，还探讨了余项的收敛性及费马近miss带来的偏差。
AI 中文摘要
考虑多项式 $f=x^N+y^N-z^N$，其中 $x、y、z$ 为正整数，$N \ge 3$ 为整数。根据费马大定理，$f$ 永远不为零，因此其倒数 $1/f$ 无奇点。我们研究该倒数的有限和：$S(m,N)=\sum_{x=1}^m\sum_{y=1}^m\sum_{z=1}^{m}\frac{1}{f}$。项 $1/f$ 可正可负，且绝对值小于1。一个关键发现是，$S(m,N)$ 可拆分为两个便于处理的部分：具有简单解析表达式的主导项 $D(m,N)$，以及更复杂但相比 $D(m,N)$ 可忽略的余项 $R(m,N)$。因此，$S(m,N)$ 几乎与 $D(m,N)$ 完全相同。$D(m,N)$ 的解析表达式为 $(2\\,m-1)\\,H_m^{(N)}$，其中 $H_m^{(N)}=\sum_{x=1}^m\frac{1}{x^N}$，当 $m$ 增大时会快速趋近于黎曼ζ函数 $\zeta(N)$。由此，原求和 $S(m,N)$ 具有简单表达式：它基本与 $m$ 呈线性关系，斜率为 $2\\,\zeta(N)$。该线性行为并非渐近结果；不同 $N$ 对应的 $S(m,N)$ 随 $m$ 变化的图像显示，从 $m=1$ 开始就是一条直线。对于 $N=3$ 的情况，$S(m,N)$ 在小区间 $8\le m\le 12$ 内略微偏离直线，这种微小偏差源于费马近miss（即 $x^3+y^3-z^3=\pm 1$，且 $z\ne x$、$z\ne y$），这些情况会在 $m=9$ 处使余项 $R(m,3)$ 产生跳变。我们对余项 $R(m,N)$ 进行了数值和解析研究。数值分析表明，当 $N\ge 4$ 时 $R(m,N)$ 收敛，但难以确定 $N=3$ 时是否收敛。基于将 $R(m,N)$ 与其柯西主值积分对比的解析研究显示，$R(m,3)$ 可能呈对数发散，同时也表明 $N\ge 4$ 时 $R(m,N)$ 收敛，与数值分析结果一致。我们在结论中讨论了一些值得未来研究的有趣问题。
英文摘要
Consider the polynomial $f=x^N+y^N-z^N$ where $x,\,y$ and $z$ are positive integers and $N \ge 3$ is an integer. By Fermat's Last Theorem, $f$ is never zero so that its reciprocal, $1/f$, has no singularities. We therefore study the finite sum of the reciprocal: $S(m,N)=\sum_{x=1}^m\sum_{y=1}^m\sum_{z=1}^{m}\frac{1}{f}$. The terms $1/f$ can be positive, negative and their magnitude is less than unity. A key observation is that $S(m,N)$ can be split into two convenient parts: a dominant contribution $D(m,N)$ that has a simple analytical expression and a remainder $R(m,N)$ which is more complicated but negligible compared to $D(m,N)$. Therefore, $S(m,N)$ is almost identical to $D(m,N)$. The analytical expression for $D(m,N)$ is $(2\,m-1)\,H_m^{(N)}$ where $H_m^{(N)}=\sum_{x=1}^m\frac{1}{x^N}$ approaches quickly the Riemann zeta function $ζ(N)$ as $m$ increases. Therefore, the original sum $S(m,N)$ has a simple expression: it is basically linear in $m$ with slope equal to $2\,ζ(N)$. Its linear behavior is not an asymptotic result; plots of $S(m,N)$ vs. $m$ for different $N$ show a straight line starting at $m=1$. $S(m,N)$ deviates slightly from a straight line over a small interval $8\le m\le 12$ for the case $N=3$. This slight deviation is due to Fermat near misses where $x^3+y^3-z^3=\pm 1$ (for $z\ne x$ and $z\ne y$); these create a jump in the remainder $R(m,3)$ at $m=9$. We make a numerical and analytical study of the remainder $R(m,N)$. From the numerical analysis, $R(m,N)$ converges for $N\ge 4$ but it was harder to tell whether $N=3$ converged. An analytical study based on a comparison of $R(m,N)$ to its Cauchy principal value integral, shows that $R(m,3)$ likely diverges logarithmically. It also shows that $R(m,N)$ converges for $N\ge 4$ in agreement with the numerical analysis. We discuss in the conclusion some interesting questions for future investigation.
Comments16 pages, 7 figures. V2: typos in conclusion fixed
