Number Theory: An Introduction to Mathematics

590 Notations

ξ,{ξ},I,φα,β(N), 448

χα,β, 449

e(t), 450

φa,b(N),Id,{x},m·x, 452

D∗N=D∗N(ξ 1 ,...,ξN),φα(N), 459

δN, 462

D∗N(x 1 ,...,xN), 464

B,A∆B,σ(A),μ(B), 465

(X,B,μ), a.e.,

∫

Xfdμ, 466

L(X,B,μ),T−^1 B, 466

λ,Ta, 472

RA, 473

p 1 ,...,pr,[a−m,...,am], 476

σ,Bp 1 ,...,pr, 477

Cia 11 ......iakk, 478

τ,B+p 1 ,...,pr,T, 479

T 1 M, 482

(X,d),A ̄, 485

g(x), 496
I 0 ,In,Jn(γ), 497
gλ(x), 498, 510
U,V, 499, 531
an,bn,M(a,b), 503
K(a,b), 504, 505
E(a,b),en, 505
P(a,b,p), 505
pn,qn, 506
c,K(a,c),cn, 507
E(a,c),K(λ),E(λ), 508
fλ(x), 510
S(t)=S(t,λ), 510, 526
E(u), 516, 530
Π(u,a), 517, 530
q,z,θ(v)=θ(v;τ), 519
θα,β(v)=θα,β(v;τ), 520
θ 00 (v),θ 01 (v),θ 10 (v),θ 11 (v), 520

θ 1 (πv,q),θ 2 (πv,q), 521

θ 3 (πv,q),θ 4 (πv,q),Q 0 , 521

snu,cnu,dnu,u=πθ 002 ( 0 )v, 525

λ(τ), 525, 531

K(τ),K′(τ), 527

Θ(u),E(K), 530

H,U,V,T,S, 531

T,T′, 533

D,D∗,D ̄, 534

F(α,β;γ;z), 537

θΛ(τ), 538

r 4 (m), 541

σ(m),σ′(m), 542

r 2 (m), 543

d 1 (m),d 3 (m),rs(m),p(n), 544

(a) 0 ,(a)n,(a)∞, 546

η(τ), 549

C,C ̄, 550

W =W(a 1 ,...,a 6 ), 552, 558

Ca,b, 553, 558

0 ,d, 553

P 1 +P 2 ,−P, 556, 583

E=E(Q),h(P), 559

hˆ(P), 560, 561

(P,Q), 563

CA,B,D,E,N, 565

E,Et,Ef, 569

∆,b 2 ,b 4 ,b 6 ,b 8 , 570

Wp,Np,cp,L(s)=L(s,W), 571

cn,N=N(W),fp,Λ(s), 572

r,E(W,Q), 572

Γ 0 (N), 573

Cn, 577

A+(n),A−(n), 578

EA,B, 579

WA,B, 580

[mn]q, 581

TheLandau order symbolsare defined in the following way: ifI=[t 0 ,∞)is a half-

line and if f,g:I →Rare real-valued functions withg(t)>0forallt∈ I,we

write

f=O(g)if there exists a constantC>0 such that|f(t)|/g(t)≤Cfor allt∈I;

f=o(g) iff(t)/g(t)→0ast→∞;

f∼g iff(t)/g(t)→1ast→∞.

Theend of a proofis denoted by
.