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 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 Definition:Category Hardy-Littlewood Circle Method Vinogradov Circle Method Characterisation of Totally Ordered Field Properties of Totally Ordered Field Gauss's Lemma (Ring Theory) 8s9j00q06ec1pixsutawbecgaerh4p4