By Jay Jorgenson

The concept of specific formulation for regularized items and sequence kinds a usual continuation of the analytic conception constructed in LNM 1564. those specific formulation can be utilized to explain the quantitative habit of varied items in analytic quantity conception and spectral conception. the current e-book bargains with different purposes bobbing up from Gaussian try capabilities, resulting in theta inversion formulation and corresponding new varieties of zeta capabilities that are Gaussian transforms of theta sequence instead of Mellin transforms, and fulfill additive practical equations. Their wide selection of functions contains the spectral conception of a wide classification of manifolds and likewise the idea of zeta capabilities in quantity idea and illustration idea. the following the hyperbolic 3-manifolds are given as an important example.

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**Extra info for Explicit Formulas for Regularized Products and Series**

**Sample text**

Let H i , . . , Hr be a finite number of functions which are meromorphic on the closure of 7r + , and let r be a function which is holomorphic on this closure. We say that a sequence {Tin} of positive 48 real n u m b e r s t e n d i n g to infinity is J - a d m i s s i b l e for {H1,. 9 9 Hr} if for any k, Hk has no zero or pole on the segment Sm = [-a + iTm,ao + a + iTm] and qS(s)H~/Hk(s) --+ 0 for s C Srn with rn --+ oo. W h e n { H 1 , . . , Hr} is the set of functions {Z, z~, ~}, we say simply that {Tin } is a d m i s s i b l e .

We define the following regions in the complex plane: ~ + = semi-infinite open rectangle bounded by the lines Re(s) = - a , Re(s) = cr0 + a, Im(s) = 0. 7E+(T) = the portion of g + below the line Im(s) = T. W e allow Z and 9 to have zeros or poles on the finite real s e g m e n t [ - a , a0 + a], but we assume that 9 and Z have no zeros or poles on the vertical edges with Re(s) = - a and Re(s) = cro + a and Im(s) > O. Let r be any function which is holomorphic on the closure of the semi-infinite rectangle 7~+, and let H be a meromorphic function on this closure.

2 will be given in the following section, and various applications of the theorem will be discussed in w 51 w P r o o f of t h e C r a m 4 r t h e o r e m . 2 follows w of [JoL 93c], which, as we shall remark below, contains one significant technical improvement over the proof of the original theorem given by Cram@r [Cr 19] for the Riemann zeta function. Choose an e > 0 sufficiently small so that Z has no zeros or poles in the open rectangle with vertices --a, --a + ie, ao + a + ie, ao + a or on the line segment [ - a + i e , ao + a + i e ] .