User:Linus44
Things to do:
Definition:Differential
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.
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
Estimation Lemma for Contour Integrals
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
Zeroes of Functions of Finite Order
Poles of Riemann Zeta Function
Riemann Zeta Has No Zeroes 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