This book creates situations in which the zeta function, or other L-functions, appear in spectral-theoretic questions. It also connects pseudo-differential analysis, or quantization theory, to analytic number theory. Series: Pseudo-Differential Operators. Num Pages: 160 pages, biography. BIC Classification: PB. Category: (UP) Postgraduate, Research & Scholarly. Dimension: 244 x 170 x 8. Weight in Grams: 267. . 2008. Paperback. . . . . Books ship from the US and Ireland.

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Bibliographic Details

Title: Quantization and Arithmetic
Publisher: Birkhauser Verlag AG
Publication Date: 2008
Language: English
ISBN 10: 3764387904
ISBN 13: 9783764387907
Binding: Soft cover
Condition: New

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Synopsis

(12) (4) Let ? be the unique even non-trivial Dirichlet character mod 12, and let ? be the unique (odd) non-trivial Dirichlet character mod 4. Consider on the line the distributions m (12) ? d (x)= ? (m)? x? , even 12 m?Z m (4) d (x)= ? (m)? x? . (1.1) odd 2 m?Z 2 i?x UnderaFouriertransformation,orundermultiplicationbythefunctionx ? e , the?rst(resp. second)ofthesedistributionsonlyundergoesmultiplicationbysome 24th (resp. 8th) root of unity. Then, consider the metaplectic representation Met, 2 a unitary representation in L (R) of the metaplectic group G, the twofold cover of the group G = SL(2,R), the de?nition of which will be recalled in Section 2: it extends as a representation in the spaceS (R) of tempered distributions. From what has just been said, if g ˜ is a point of G lying above g? G,andif d = d even g ˜ ?1 or d , the distribution d =Met(g˜ )d only depends on the class of g in the odd homogeneousspace?\G=SL(2,Z)\G,uptomultiplicationbysomephasefactor, by which we mean any complex number of absolute value 1 depending only on g ˜. On the other hand, a function u?S(R) is perfectly characterized by its scalar g ˜ productsagainstthedistributionsd ,sinceonehasforsomeappropriateconstants C , C the identities 0 1 g ˜ 2 2 | d ,u | dg = C u if u is even, 2 0 even L (R) ?\G

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