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Things to do:


Let $\struct {E, \norm {\, \cdot \,}_E}$, $\struct {F, \norm {\, \cdot \,}_F}$ 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 $\d f_a \in \map \LL {E, F}$ such that

$\ds \lim_{h \mathop \to 0} \norm {\map f {a + h} - \map f a - \d f_a \cdot h}_F \norm h_E^{-1} = 0$

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

We say that $f$ is continuously differentiable if and only if:

$\d f : \struct {U, \norm {\, \cdot \,}_E} \to \struct {\map \LL {E, F}, \norm {\, \cdot \,}_{\map \LL {E, F} } }$
$: a \mapsto \d f_a$

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.


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:

Bézout domain

Definition:Unique Factorization Domain:

etc...needs organizing into something more standardized

The rest

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 Gamma Function

Properties of Gamma Function

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)

Open Sets in Real Number Line

Equivalence of Definitions of Riemann Zeta Function


Hardy-Littlewood Circle Method

Vinogradov Circle Method

Characterisation of Totally Ordered Field

Properties of Totally Ordered Field

Gauss's Lemma (Ring Theory)