\noindent
If the positive integer $x$ is not a divisor of $m$, the terms of this new
sum are zero except when $m$ $-$ $k$ = $m$ mod $x$, when we have $a_{m-k}(x)-
a_{m-k-1}(x) = 1$. On the other hand if \testit{x} is a divisor of $x$, the
only nonvanishing terms occurs for $m - k = x$, when we have $a_{m-k}(x)-
a_{m-k-1}(x) = 0 - (x - 1) $. Hence the sum is $1-x$[$x$ divides $m$]. Libri
obtained his complicated formula by a less direct method,
applying Newton’s identities to compute the sum of the mth powers of the
roots of the equation $t^{x-1}+t^{x-2}+\cdot\cdot\cdot+1=0$.)
\newline``
并且在此部分中它也给出了同样的错误。
\indent Some of Libri’s papers are still well remembered, but [32] and [33]
are not. I found no mention of them in $Science\; Citation\; Index$, after
searching through all years of that index available in our library (1955 to
date). However, the paper [33] did produce several ripples in mathematical
waters when it originally appeared, because it stirred up a controversy about
whether $0^0$ is defined. Most mathematicians agreed that $0^0 = 1$, but
Cauchy [5, page 70] had listed $0^0$ together with other expression like
$0/0$ and $\infty -\infty$ in a table of undefined forms. Libri’s
justification for the equation $0^0 = 1$ was far from convincing, and a
commentator who signed his name simply y “S” rose tothe attack [45]. August
M¨obius [36] defended Libri, by presenting his former professor’s reason for
believing that $0^0 = 1$ (basically a proof that $\lim_{x \rightarrow 0+} x^x
= 1).$ M\"obiuss also went further and presented a supposed proof that
$\lim_{x \rightarrow 0+} f(x)^{g(x)} = 1$ whenever $\lim_{x \rightarrow 0+}
f(x) = {\space}$\lim_{x \rightarrow 0+} g(x) = 0.$ Of course “S” then asked
[3] whether M\"obius knew about functions such as $f(x) = e^{-1/x} $ and $
g(x) = x.$ (And paper [36] was quietly omitted from the historical record
when the collected works of M\"obius were ultimately published.) The debate
stopped there, apparently with the conclusion that
$0^0$ should be undefined.
最后，我得到了正确的方程式，因为生成了 pdf，但它一直给我这两个实例相同的错误。我对 latex 还很陌生，所以我不太确定这些错误。谢谢你的帮助！
答案1
欢迎来到 TeX.SE！正如 @Mico 所指出的，有一个\testit可能应该是\textit。实际上，在您使用此命令的地方，它$x$无论如何都应该是 ，并且有一个地方$应该是\textit：\textit{Science Citation Index}。除了修复它之外，我还必须删除“Newton”、“Libri” 后面的几个特殊字符、符号¨等，以使您的代码可编译。最后，我将其修复$m$ $-$ $k$ = $m$ mod $x$为$m-k=m\mod x$，这就是我加载 的原因amsmath，我将其替换\cdot\cdot\cdot为\cdots。
\documentclass[a4paper,12pt]{article}
\usepackage{amsmath}
\begin{document}
\noindent
If the positive integer $x$ is not a divisor of $m$, the terms of this new
sum are zero except when $m-k=m\mod x$, when we have $a_{m-k}(x)-
a_{m-k-1}(x) = 1$. On the other hand if $x$ is a divisor of $x$, the
only nonvanishing terms occurs for $m - k = x$, when we have $a_{m-k}(x)-
a_{m-k-1}(x) = 0 - (x - 1) $. Hence the sum is $1-x$ [$x$ divides $m$]. Libri
obtained his complicated formula by a less direct method,
applying Newton's identities to compute the sum of the mth powers of the
roots of the equation $t^{x-1}+t^{x-2}+\cdots+1=0$.
Some of Libri's papers are still well remembered, but [32] and [33]
are not. I found no mention of them in \textit{Science Citation Index}, after
searching through all years of that index available in our library (1955 to
date). However, the paper [33] did produce several ripples in mathematical
waters when it originally appeared, because it stirred up a controversy about
whether $0^0$ is defined. Most mathematicians agreed that $0^0 = 1$, but
Cauchy [5, page 70] had listed $0^0$ together with other expression like
$0/0$ and $\infty -\infty$ in a table of undefined forms. Libri's
justification for the equation $0^0 = 1$ was far from convincing, and a
commentator who signed his name simply ``S'' rose tothe attack [45]. August
M\"obius [36] defended Libri, by presenting his former professor's reason for
believing that $0^0 = 1$ (basically a proof that $\lim_{x \rightarrow 0+} x^x
= 1).$ M\"obius also went further and presented a supposed proof that
$\lim_{x \rightarrow 0+} f(x)^{g(x)} = 1$ whenever $\lim_{x \rightarrow 0+}
f(x) = \lim_{x \rightarrow 0+} g(x) = 0.$ Of course``S'' then asked
[3] whether M\"obius knew about functions such as $f(x) = e^{-1/x} $ and $
g(x) = x.$ (And paper [36] was quietly omitted from the historical record
when the collected works of M\"obius were ultimately published.) The debate
stopped there, apparently with the conclusion that
$0^0$ should be undefined.
\end{document}
我希望已经找到了大多数问题，并且您不要像[44]在真实文档中那样对参考编号进行硬编码，而是使用bibtex、biblatex或biber类似的东西。