LEADER 04041nam 22005895 450 001 9910799233003321 005 20240102172854.0 010 $a981-9909-10-4 024 7 $a10.1007/978-981-99-0910-0 035 $a(CKB)29526685300041 035 $a(DE-He213)978-981-99-0910-0 035 $a(MiAaPQ)EBC31094165 035 $a(Au-PeEL)EBL31094165 035 $a(EXLCZ)9929526685300041 100 $a20240102d2023 u| 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 14$aThe Theory of Zeta-Functions of Root Systems$b[electronic resource] /$fby Yasushi Komori, Kohji Matsumoto, Hirofumi Tsumura 205 $a1st ed. 2023. 210 1$aSingapore :$cSpringer Nature Singapore :$cImprint: Springer,$d2023. 215 $a1 online resource (IX, 414 p. 13 illus.) 225 1 $aSpringer Monographs in Mathematics,$x2196-9922 311 08$a9789819909094 327 $aIntroduction -- Fundamentals of the theory of Lie algebras and root systems -- Definitions and examples -- Values at positive even integer points -- Convex polytopes and the rationality -- The recursive structure -- The meromorphic continuation -- Functional relations (I) -- Functional relations (II) -- PoincarŽe polynomials and values at integer points -- The case of the exceptional algebra G2 -- Applications to multiple zeta values (I) -- Applications to multiple zeta values (II) -- L-functions -- Zeta-functions of Lie groups -- Lattice sums of hyperplane arrangements -- Miscellaneous results. 330 $aThe contents of this book was created by the authors as a simultaneous generalization of Witten zeta-functions, Mordell?Tornheim multiple zeta-functions, and Euler?Zagier multiple zeta-functions. Zeta-functions of root systems are defined by certain multiple series, given in terms of root systems. Therefore, they intrinsically have the action of associated Weyl groups. The exposition begins with a brief introduction to the theory of Lie algebras and root systems and then provides the definition of zeta-functions of root systems, explicit examples associated with various simple Lie algebras, meromorphic continuation and recursive analytic structure described by Dynkin diagrams, special values at integer points, functional relations, and the background given by the action of Weyl groups. In particular, an explicit form of Witten?s volume formula is provided. It is shown that various relations among special values of Euler?Zagier multiple zeta-functions?which usually are called multiple zeta values (MZVs) and are quite important in connection with Zagier?s conjecture?are just special cases of various functional relations among zeta-functions of root systems. The authors further provide other applications to the theory of MZVs and also introduce generalizations with Dirichlet characters, and with certain congruence conditions. The book concludes with a brief description of other relevant topics. 410 0$aSpringer Monographs in Mathematics,$x2196-9922 606 $aNumber theory 606 $aGroup theory 606 $aFunctions of complex variables 606 $aNumber Theory 606 $aGroup Theory and Generalizations 606 $aFunctions of a Complex Variable 615 0$aNumber theory. 615 0$aGroup theory. 615 0$aFunctions of complex variables. 615 14$aNumber Theory. 615 24$aGroup Theory and Generalizations. 615 24$aFunctions of a Complex Variable. 676 $a512.7 700 $aKomori$b Yasushi$4aut$4http://id.loc.gov/vocabulary/relators/aut$01587639 702 $aMatsumoto$b Kohji$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aTsumura$b Hirofumi$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910799233003321 996 $aThe Theory of Zeta-Functions of Root Systems$93875878 997 $aUNINA