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Download Basic Analysis of Regularized Series and Products by Jay Jorgenson PDF

By Jay Jorgenson

Analytic quantity conception and a part of the spectral conception of operators (differential, pseudo-differential, elliptic, etc.) are being merged less than amore basic analytic idea of regularized items of sure sequences fulfilling a couple of simple axioms. the main simple examples encompass the series of traditional numbers, the series of zeros with optimistic imaginary a part of the Riemann zeta functionality, and the series of eigenvalues, say of a favorable Laplacian on a compact or convinced situations of non-compact manifolds. The ensuing thought is acceptable to ergodic thought and dynamical structures; to the zeta and L-functions of quantity conception or illustration conception and modular kinds; to Selberg-like zeta capabilities; andto the idea of regularized determinants time-honored in physics and different elements of arithmetic. apart from proposing a scientific account of greatly scattered effects, the idea additionally presents new effects. One a part of the idea bargains with complicated analytic houses, and one other half bargains with Fourier research. regular examples are given. This LNM presents simple effects that are and should be utilized in additional papers, beginning with a common formula of Cram r's theorem and particular formulation. The exposition is self-contained (except for far-reaching examples), requiring basically normal wisdom of analysis.

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Example text

R e m a r k 1. A s s u m e L can be w r i t t e n as the disjoint u n i o n of the sequences L ~ a n d L" w h e r e L ~ satisfies the convergence conditions D I R 1, D I R 2 a n d D I R 3, a n d Oc,(t) satisfies the a s y m p t o t i c 35 conditions A S 1, A S 2 and A S 3. T h e n L" and OL,,(t) necessarily satisfy these conditions and 0(~) = 0L,(~) + 0L,,(~). F r o m this, we i m m e d i a t e l y have ~(~, z) = ~L,(~, z) + ~L,,(~, ~) and DL(z)=DL,(Z)DL,,(z). A p a r t i c u l a r example of such a d e c o m p o s i t i o n is the case w h e n L' is a finite subset of L, in which case DL,(Z) = 1--[[(z + Ak)e'~]; see R e m a r k 2 below.

Proof. Let us first show how A S 3 follows. Directly from D I R 2 and D I R 3 we have, for some constants Cl and c2, the inequalities s I O(t) -- ONO(t) < k=Nake -)~kt <- Z la'~f~-I~~ k=N OO _< c, ~ lakl~~ -~'~''. k=N Note t h a t for any x _> 0, there is a constant c = c(~r0 + (7 1 ) such that X a~ -x ~ C. 27 Let us apply this inequality to x to obtain, for any t > 0, = c2[Ak[t and t h e n sum for k > N oo O(t) -QNO(t) <-- C1 E ]'~k]a~ k=N (7) Cl" C(c2t)--a~ oo " Z [/~kl-trl" k=N Now if we let O~ = 0r0 AV (71 and oo c = c,4c~)-~ Z I:,,,I -~, , k=l t h e n (7) becomes O(t) - QNO(t) <_C/t ~, which establishes the a s y m p t o t i c condition A S 3.

To begin, we will study the asymptotics of oo ~(s,z) = --] O(t)e-~tt ~dtt J for Re(z) -~ ec. 0 PqO be as in A S 2. ~(s,z) : Jl(s,z) + J2(~,z) + J3(s,z) Fix Re(q) > 0 and, as before, let Let us write where 1 (1) Jl(s,z) = f(O(t)- PqO(t)) -z, t dtt ' 0 (2) J2(s,z) = ff O(t)e-Ztt ~dt t' 1 1 (3) J3(s,z) = / P'O(t)e-Ztt~ dtt 0 Recall that m(q)=maxdegBp for Re(p)=Re(q). The terms in (1) and (2) are handled in the two following lemmas. 1. fix q with x = Re(z) a n d Jl(O,z) = O(x-Re(q)(logx)m(q)) Proof.

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