User:Linus44

Things to do:

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
GCD domain:
 * Equivalent Definitions of a GCD Domain
 * Properties of GCD Domains

Bézout domain
 * Equivalent Definitions of a Bézout domain
 * Properties of Bézout Domains

Definition:Unique Factorization Domain:
 * Equivalent Definitions of a Unique Factorization Domain
 * Properties of Unique Factorization Domains

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

Poisson Summation Formula

Definition:Schwarz Function

Harmonic Properties of Schwarz Functions

Hadamard Factorisation Theorem

Jensen's Formula

Definition:Completed Riemann Zeta Function

Estimation Lemma

Uniform Limit of Analytic Functions is Analytic

Poles of the Gamma Function

Properties of the Gamma Function

Stirling's Formula for the Gamma Function

Definition:Order of an 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-Functions

L-functions do not Vanish at One

Logarithm of Dirichlet L-Functions

Dirichlet's Theorem on Arithmetic Progressions

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)

Open Sets in Real Number Line

Equivalence of Riemann Zeta Function Definitions

Definition:Category

Hardy-Littlewood Circle Method

Vinogradov Circle Method

Characterisation of Totally Ordered Fields

Properties of Totally Ordered Fields

Gauss's Lemma (Ring Theory)