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Call a set $A\subseteq[a,b]$ thick in the interval $[a,b]$, if for all $t\in(a,b)$ we have $|A\cap[a,t]|>0.6(t-a)$ and $|A\cap[t,b]|>0.6(b-t)$. The pigeon hole principle implies that if $A, B$ are thi …

The number of totient divisors of $n$ is $d(n-1)-d((n-1, \varphi(n))$. As $n$ gets large, then almost all $n$ have the property that $\varphi(n)$ is divisble by all small primes. The average number of …

Make Weyl-van der Corput estimates explicit: Let $f:[a,b]\rightarrow\mathbb{R}$ be a somewhat smooth function, and assume that you have some bounds on certain derivatives of $f$. Then give an upper b …

Write the function $x\mapsto\{x\}-\frac{1}{2}$ as a Fourier series, and approximate this series by a smoothed finite sum using Vaaler's lemma. You obtain something of the form $$ \sum_{a<n\leq b}\left …

The number of necklaces of size $p$ is $\frac{a^p}{p}+\mathcal{O}(a^{p/2})$, hence $$ \prod_{p=1}^nN(p,a)=\frac{a^{n(n+1)/2}}{n!}\prod_{p=1}^n\left(1+\mathcal{O}(a^{-p/2})\right) = \left(c+\mathcal{O} …

Which formula to prefer depends mainly on what you want to do with it. Do you need high precision, or do you have to do complicated things with the approximation? The expansion in Hermite-polynomials …