Number Theory: An Introduction to Mathematics

406 X A Character Study Proof Since fis holomorphic atβ, we can chooseδ>0sothatfis holomorphic in the disc|s−(β+δ)|< 2 δ. ...

3 Proof of the Prime Number Theorem for Arithmetic Progressions 407 −f′(s)/f(s) =− 2 ζ′(s)/ζ(s)−L′(s+iα,χ)/L(s+iα,χ)−L′(s−iα,χ)/ ...

408 X A Character Study Proposition 8 ∑ n≤xχ^1 (n)Λ(n)∼x, ∑ n≤xχ(n)Λ(n)=o(x)ifχ=χ^1. Proof For any Dirichlet characterχ, put g( ...

3 Proof of the Prime Number Theorem for Arithmetic Progressions 409 ψ(x;m,a)=x/φ(m)+O(x/logαx), π(x;m,a)=Li(x)/φ(m)+O(x/logαx), ...

410 X A Character Study 4 Representations of Arbitrary Finite Groups The problem of extending the character theory of finite abe ...

4 Representations of Arbitrary Finite Groups 411 It is easily verified that ifs→A(s)is a matrix representation of degreenof a gr ...

412 X A Character Study Proof We give a constructive proof due to Schur. Lets→A(s),where A(s)= ( P(s) Q(s) 0 R(s) ) , be a reduc ...

4 Representations of Arbitrary Finite Groups 413 is also a positive definite inner product onVand that it is invariant underG,i. ...

414 X A Character Study 5 Characters of Arbitrary Finite Groups.......................... By definition, thetraceof ann×nmatrixA ...

5 Characters of Arbitrary Finite Groups 415 is a diagonal matrix. Moreover, since T−^1 SkT=diag[ω 1 k,...,ωkn], ω 1 ,...,ωnare a ...

416 X A Character Study For suppose there existλ(μ)ij ∈Csuch that ∑ i,j,μ λ(μ)ij αij(μ)(s)=0foreverys∈G. Multiplying byα(klv)(s− ...

5 Characters of Arbitrary Finite Groups 417 Sinceχv(e)=nvis the degree of thev-th irreducible representation, it follows from (7 ...

418 X A Character Study i.e. if and only ifr=g, it follows thata finite group is abelian if and only if every irreducible repres ...

6 Induced Representations and Examples 419 by (10). IfρR(C)is the matrix representingCin the regular representation, it fol- low ...

420 X A Character Study With respect to a given basis ofVletA(t)now denote the matrix representing t∈Hand putA(s)=Oifs∈G\H. If o ...

6 Induced Representations and Examples 421 Proof Letχdenote the character of the representationρofGandψthe character of the repr ...

422 X A Character Study Consider now the induced representationψ ̃iofG.SinceHis a normal subgroup, it follows from (12) that ψ ̃ ...

6 Induced Representations and Examples 423 1 =g−^1 ∑ s∈G ψ ̃i(s)χk(s−^1 ) =g−^1 ∑ s∈H ψ ̃i(s)χk(s−^1 ) =g−^1 ∑ s∈H ψ ̃i(s)χk(s−^ ...

424 X A Character Study elements (12),(13),(23) of order 2, andC 3 containing the two elements (123),(132) of order 3. The irred ...

7 Applications 425 self-conjugate characterψ 4 ofA 4 yields two irreducible charactersχ 4 ,χ 5 ofS 4 of degree 3. The rows of th ...