# User:Linus44

Things to do:

## Contents

## Definition:Differential

Let $(E, \| \cdot \|_E)$, $(F, \| \cdot \|_F)$ be normed vector spaces.

Let $U \subseteq E$ be an open set.

Let $f : U \to F$ be a mapping.

Let $a \in U$ be an element of $U$.

Then $f$ is **differentiable** at $a$ if there exists a continuous and linear map $df_a \in \mathcal L(E,F)$ such that

- $\displaystyle \lim_{h \to 0} \| f(a+h) - f(a) - df_a \cdot h \|_F \| h \|_E^{-1} = 0$

Then $df_a$ is called the **differental** or the **tangent map** of $f$ at $a$.

We say that $f$ is **continuously differentiable** if:

- $\displaystyle df : (U, \| \cdot \|_E) \to \mathcal (L(E,F),\| \cdot \|_{L(E,F)})$
- $\displaystyle \ : a \mapsto df_a$

is continuous.

If $E = \R^n$, this is true iff the first order partial derivatives of $f$ exist and are continuous.

## Induced polynomial homomorphism

Even this needs serious thought if it's to be any good.

## Permutations

Definition:Cyclic Permutation $k$ well defined. Add canonicality.

Incorrect: Definition:Permutation on n Letters/Cycle Notation permutation/cycle confusion? Also $\rho$ should be $\pi$ for consistency.

Then here: Equality of Cycles

## Rings, properties, equivalent definitions

Definition:Unique Factorization Domain:

etc...needs organizing into something more standardized

## The rest

- Vinogradov's Theorem: Pick a method of proof and think of a good structure for it.

- Figure out conditions for the "function-form epimorphism" to have trivial kernel (see Epimorphism from Polynomial Forms to Polynomial Functions and especially Equality of Polynomials)

### Proof of prime number theorem and Siegel Walfiz

See also stuff on Dirichlet's Theorem

Harmonic Properties of Schwarz Functions

Hadamard Factorisation Theorem

Definition:Completed Riemann Zeta Function

Uniform Limit of Analytic Functions is Analytic

Stirling's Formula for Gamma Function

Definition:Order of Entire Function

Completed Riemann Zeta Function has Order One

Product Equation for Riemann Zeta Function

Zeros of Functions of Finite Order

Poles of Riemann Zeta Function

Riemann Zeta Has No Zeros With Real Part One

Unsymmetric Functional Equation for Riemann Zeta Function

### Dirichlet: Finished, but check for errors

Analytic Continuation of Dirichlet L-Function

L-Function does not Vanish at One

Logarithm of Dirichlet L-Functions

Dirichlet's Theorem on Arithmetic Sequences

Dirichlet L-Function from Trivial Character

Convergence of Dirichlet Series with Bounded Coefficients

Convergence of Dirichlet Series with Bounded Partial Sums

Definition:Dirichlet L-function

Definition:Dirichlet Character

Definition:Completed Dirichlet L-Function

Functional Equation for Dirichlet L-Functions

Orthogonality Relations for Characters

### Unfinished Pages

Completeness Criterion (Metric Spaces)

Equivalence of Definitions of Riemann Zeta Function

Hardy-Littlewood Circle Method

Characterisation of Totally Ordered Field