An immersion $\varphi :U\to {\mathbb{R}}^3$ has differential of rank $2$, equivalently linearly independent ${\varphi }_u,{\varphi }_v$. It has isothermal coordinates when its first fundamental form has $E=G>0$, $F=0$. A minimal surface has zero mean curvature.

For an isothermal immersion, differentiation of $E=G$ and $F=0$ givesand similarly its scalar product with ${\varphi }_v$ is zero. Its normal component is $e+g$. The given mean curvature formula reduces to $H=(e+g)/(2E)$, soThus an isothermal immersion is minimal exactly when each coordinate is a harmonic function.

An explicit example is the Enneper surface, parametrized on ${\mathbb{R}}^2$ byEvery coordinate has zero Laplacian. Direct differentiation gives ${\varphi }_u\cdot {\varphi }_v=0$ and $|{\varphi }_u{|}^2=|{\varphi }_v{|}^2=(1+u^2+v^2{)}^2>0$. Thus it is an isothermal immersion and is minimal everywhere.

It is not an open subset of any of the three listed surfaces. The parameter points $(\sqrt{3},0)$ and $(-\sqrt{3},0)$ both map to $(0,0,3)$, but their tangent planes differ: ${\varphi }_u=(-2,0,\pm 2\sqrt{3})$ while ${\varphi }_v=(0,4,0)$. A plane, catenoid, or helicoid is an embedded smooth surface with a unique tangent plane at each point. If our whole image lay in any one of them, both rank-$2$ tangent planes at this point would have to equal its tangent plane, a contradiction.

The Euler totient function counts integers $1\le a\le n$ with $\gcd (a,n)=1$. The Möbius function has $\mu (1)=1$, is zero when a prime square divides $n$, and equals $(-1{)}^r$ when $n$ is a product of $r$ distinct primes. Consequentlybecause for $n>1$ the sum over squarefree divisors is $(1-1{)}^r$.

The Möbius inversion formula is $g(n)={\sum }_{d\mid n}\mu (d)f(n/d)$. Substituting the divisor sum defining $f$ and collecting the coefficient of $g(e)$ givesThis proof uses only finite sums, so no convergence hypothesis is needed.

Classifying $1,\dots ,n$ by their greatest common divisor with $n$ gives ${\sum }_{d\mid n}\phi (d)=n$: the class with greatest common divisor $n/d$ contains exactly $\phi (d)$ integers. Applying the Möbius inversion formula yieldsThe Riemann zeta function is $\zeta (s)={\sum }_{n\ge 1}{n}^{-s}$ for $\mathrm{Re}s>1$. Absolute convergence justifies multiplication of Dirichlet series. The divisor identity for the Möbius function gives $\sum \mu (n)n^{-s}=1/\zeta (s)$ there. For $\mathrm{Re}s>2$ both factors below converge absolutely, and their Dirichlet convolution is the Euler totient function:

A covering map is a continuous surjection $p:\tilde{X}\to X$ for which each point has an open neighbourhood $U$ with $p^{-1}(U)$ a disjoint union of open sets, each mapped homeomorphically onto $U$. A connected simply connected covering space is a universal cover.

There is a necessary connectedness qualification in the printed conclusion. Let $A_0$ be the path component of $a_0$ in $A$. Then $p(\tilde{A})=A_0$: one inclusion follows by projecting paths, and the other by lifting a path in $A_0$ from $a_0$. Thus the asserted surjection onto all of $A$ requires $A$ to be path connected. Without it, take $p$ to be the identity of $\mathbb{R}$ and $A$ to be two disjoint intervals; a selected component cannot cover both.

To prove the assertion about a component of a pulled-back universal cover, choose an evenly covered neighbourhood $U$ in $X$. Local path connectedness of $A$ supplies a path-connected relatively open neighbourhood $V\subseteq U\cap A_0$. Each inverse sheet over $V$ is path connected, so it lies wholly in $\tilde{A}$ or is disjoint from it. These sheets evenly cover $V$ under $p{|}_{\tilde{A}}$. Therefore $p{|}_{\tilde{A}}:\tilde{A}\to A_0$ is a covering map.

Under the classification of connected covering spaces, its subgroup consists of based loops in $A_0$ whose lifts close at ${\tilde{a}}_0$. A loop viewed in $X$ closes in the universal cover exactly when its class in ${\pi }_1(X,a_0)$ is trivial. Hence the subgroup isHere ${\pi }_1(A,a_0)={\pi }_1(A_0,a_0)$, so the subgroup formula itself needs no change. Injectivity of the covering's induced fundamental-group map is the usual homotopy-lifting property.