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.

(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

Consists entirely of new material Contents is related to the book Automorphic pseudodifferential analysis and higher-level Weyl calculi (appeared in Progress in Mathematics) by the same author: one-dimensional pseudo-differential analysis is considered in both volumes; however, the results are disjoint: the former book dealt with automorphic symbols, while in the present one, it is the distributions to which the operators under consideration are applied that carry the arithmetic; in this way, L-functions, rather than non-holomorphic modular forms, have the dominant role Includes supplementary material: sn.pub/extras

Inhalt
Weyl Calculus and Arithmetic.- Quantization.- Quantization and Modular Forms.- Back to the Weyl Calculus.