By Jay Jorgenson

ISBN-10: 0387586733

ISBN-13: 9780387586731

ISBN-10: 3540586733

ISBN-13: 9783540586739

The concept of specific formulation for regularized items and sequence kinds a typical continuation of the analytic idea constructed in LNM 1564. those particular formulation can be utilized to explain the quantitative habit of varied gadgets in analytic quantity idea and spectral concept. the current e-book offers with different functions coming up from Gaussian try services, resulting in theta inversion formulation and corresponding new varieties of zeta features that are Gaussian transforms of theta sequence instead of Mellin transforms, and fulfill additive useful equations. Their wide selection of purposes comprises the spectral thought of a extensive type of manifolds and in addition the speculation of zeta services in quantity conception and illustration thought. the following the hyperbolic 3-manifolds are given as an important example.

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As above, let a=-Re(p0) and /3=-re(p0). O ) OuI = 0 CJ 1 T ~ (log T ) - # =0 dx (c2(x(logx)#) 1/'~ x-('~+l)/~(log _ x)-#(n+l)/adx I . w~nT(2T){~"~, 0) ---- O ( x-(n+l)/a+l(log x)-#(r~+l)/adx 4 c l T ~ (log T ) - # o ((~O(~o~)~) -(~247176247 : o (r~247 ~)-~) -- o (T--o(~o)-(o§ T)~(~0)) _- ) ~(o§ To finish one integrates n-times. An extra power of log T occurs precisely in the case when Re(p0) E Z<0. 3 are for T --~ ~ . To estimate the remaining term in (1), namely the finite sum >~ I~kl<2T ak u+w+Ak' it is necessary to choose carefully a sequence Tn.

In the spectral case, we apply the Karamata theorem to (5) with the family of test functions described above and a=-Re(p0) and fl=-rn(p0), from which we obtain the estimate (8) N ( A ) - - # { k : IAkl _< A} ~ C'A~(log A) -~ for some positive constant C I. This estimate gives the relation k = X(Ak) ~ C'lAkl~(loglAkl)-~, as asserted in the statement of the lemma. Then k 1/'~ ~ C'l/~[Ak I (log lakl) and logk ~ a . log I kl. Hence, for some positive constant C, we obtain the asymptotic relation IAkl ~ c ' - l / ~ 1/~ (log IAkl) ~ / ~ ~ C]~ 1/~ (log k) ~/'~ , which completes the proof of the lemma.

W~nT(2T){~"~, 0) ---- O ( x-(n+l)/a+l(log x)-#(r~+l)/adx 4 c l T ~ (log T ) - # o ((~O(~o~)~) -(~247176247 : o (r~247 ~)-~) -- o (T--o(~o)-(o§ T)~(~0)) _- ) ~(o§ To finish one integrates n-times. An extra power of log T occurs precisely in the case when Re(p0) E Z<0. 3 are for T --~ ~ . To estimate the remaining term in (1), namely the finite sum >~ I~kl<2T ak u+w+Ak' it is necessary to choose carefully a sequence Tn. Roughly speaking, the sequence must be as far from any --~k as possible. The following lemma makes this statement precise.

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Explicit Formulas for Regularized Products and Series by Jay Jorgenson


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