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For $K=\mathbb{Q}(\sqrt{5})$,Writing $\varphi =(1+\sqrt{5})/2$, the full unit group is

An Euler product factors a Dirichlet series into local factors indexed by prime numbers. For a multiplicative arithmetic function $f$ and in a half-plane of absolute convergence,

The Dirichlet hyperbola method turns a long summation over a Dirichlet convolution into two short ones by covering the lattice points below a hyperbola with two overlapping strips. Its simplest application is an exact count: for a positive integer $N$, every positive integer divisor $a$ of $n\le N$ corresponds to the ordered pair $(a,n/a)$. Thus the divisor summatory function counts precisely the ordered pairs of positive integers with $ab\le N$:Here $\tau$ is the divisor function, and the floor function gives the number of allowed values of $b$ for each $a$. The axes are excluded, while points on $ab=N$ are included.

Set $r=\lfloor \sqrt{N}\rfloor$, using the floor function and the positive square root. Every counted pair has $a\le r$ or $b\le r$. Otherwise $ab\ge (r+1{)}^2>N$. The first strip contains ${\sum }_{a=1}^r\lfloor N/a\rfloor$ points; swapping the two coordinates gives the same count for the second. Their overlap contains all $r^2$ pairs with $1\le a,b\le r$, since $r^2\le N$. The inclusion-exclusion principle therefore gives the exact identityThe subtraction is for the whole overlap square, rather than only the diagonal. The same Dirichlet hyperbola method identity holds for real $x\ge 1$, with $D(x)={\sum }_{n\le x}\tau (n)$ and $r=\lfloor \sqrt{x}\rfloor$. ^[ref]

The functional equation may be written

For a primitive nonprincipal Dirichlet character with parity $a$, the Gamma function poles in its completed Dirichlet L-function force zeros of $L(s,\chi )$ at $s=-a-2m$, $m=0,1,\dots$. The functional equation and nonvanishing of nonprincipal Dirichlet L-functions at one show that these zeros are simple and cancel the poles, rather than remaining zeros of the completion. For the Dirichlet beta function, the odd-parity examples are $\beta (-1)=\beta (-3)=\cdots =0$. The primitive principal conductor-one case is different: its zeta trivial zeros start at $-2$ and the completion has poles at zero and one.

Hankel residue extraction gives $\beta (0)=1/2$, $\beta (-2)=-1/2$, and values at other nonpositive integers from the Taylor coefficients of $1/(2\cosh t)$.

The adjacency matrix of a finite directed graph has entry $A_{ij}$ equal to the number of directed edges from vertex $i$ to vertex $j$. The entry $(A^n{)}_{ij}$ counts length-$n$ directed walks from $i$ to $j$, so $\mathrm{tr}(A^n)$ counts pointed closed directed walks of length $n$.

A directed walk is a sequence of vertices in which each consecutive pair is joined by a directed edge in the direction traversed.

A directed edge $u\to v$ has an initial vertex $u$ and a terminal vertex $v$.

For suitable convex integral functionals, ordered boundary values of two minimizers imply the same ordering in the domain. Affine minimizers can therefore serve as upper and lower barriers.

Boundary data $g$ satisfy the bounded slope condition with constant $K$ when every boundary point admits affine lower and upper supporting functions of Lipschitz constant at most $K$. Comparison with these barriers gives a global Lipschitz bound for convex variational problems.

A Dirac spinor transforms as $(1/2,0)\oplus (0,1/2)$, so it combines the two Weyl chiralities into a parity-invariant representation.

A left- or right-handed Weyl spinor transforms in the Lorentz representation $(1/2,0)$ or $(0,1/2)$. Parity exchanges the two chiralities.

The Dirac equation is the relativistic spin-one-half field equation $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$.

For nonzero $a$, $\delta (ax)=\delta (x)/|a|$ in the distributional sense.

For every test function $\phi$,so $H^{\prime }=\delta$ as distributions.

A reaction-diffusion system combines local reaction kinetics with spatial diffusion,

A nonlinear diffusion equation lets diffusivity depend on the evolving field; the porous-medium equation is a standard example.

The constant-coefficient one-dimensional advection-diffusion equation isTranslation to coordinates moving at speed $a$ reduces it to the heat equation.

The heat equationmodels diffusion with constant diffusivity $D$.