I was thinking, specifically, of this paper, in which Zagier offers a proof of the PNT, inspired by a paper of Newman's, the Cliffs Notes version of which would be that, first, it's fairly easy to show that $$\left|\int_1^{\infty}\frac{\vartheta(x) - x}{x^2}dx\right| < \infty \implies \vartheta(...

Sketch the curve $\mathrm{f}(x)=x^{3}+Ax^{2}+B$ first in the case $A>0$ and $B>0$, and then in the case $A<0$ and $B>0.$ Show that the equation $x^{3}+ax^{2}+b=0$, where $a$ and $b$ are real, will have three distinct real roots if $27b^{2}+3a^{3}b<0,$ but will have fewer than three if $27b^{...

Define a monomial function $f: G \to G$ to be a function that can be written in the form $f(x) = g x g'$ for some $g, g' \in G$. Let $G$ be a finite group and consider the set of $S$ of all monomial functions from $G$ to $G$ that take one argument. For instance $(f(x) = abxcd) \in S$. Now if $...

What are the criteria such that a function $f(t)$ can serve as the correction in an iteration function of the form $g (t) = t - \lambda f (t)$ where $\lambda$ is some relaxation factor? It is almost reminiscent of Newton's iteration, without the derivative. for instance, if $f(t)=sin(t)$ then $g...