MathOverflow

A result of Selberg (A. Selberg. On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvid., 47(6):87–105, 1943) says essentially
$$\int _1^x(\psi (t+h)-\psi (t)-h)^2dt\ll xh\hspace {15mm}\psi (x):=\sum _{n<x}\Lambda (n).$$...

Given an increasing function $G \colon [0, \infty) \to [0, \infty)$ with $G(x) \rightarrow \infty$ for $x \to \infty$, is there a (strictly) decreasing (differentiable) function $f \colon [0, \infty) \to [0, \infty)$ with $\int_{0}^{x} f(t) \mathrm{d...

Let $G=\mathrm{SL}_2(\mathbb C)$, $V$ its standard representation, and $V_m=\operatorname{Sym}^m(V)$ with $m\equiv 2 \pmod 4$. It is classical that$$\Lambda^2 V_m \;\cong\; \bigoplus_{\substack{1\le r\le m\\ r\ \text{odd}}} V_{\,2m-2r}$$is multiplicity...

Background:
Let $G$ be a finite group. Fix a prime $p$. We say that $H\subseteq G$ is strongly $p$-embedded if:
$p$ divides $|H|$;