LEADER 03908nam 22004695 450 001 9910999786003321 005 20250427130146.0 010 $a3-031-85145-5 024 7 $a10.1007/978-3-031-85145-2 035 $a(MiAaPQ)EBC32029919 035 $a(Au-PeEL)EBL32029919 035 $a(CKB)38602799300041 035 $a(DE-He213)978-3-031-85145-2 035 $a(EXLCZ)9938602799300041 100 $a20250427d2025 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aL-Functions $eAn Elementary Introduction /$fby Davide Lombardo 205 $a1st ed. 2025. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2025. 215 $a1 online resource (495 pages) 225 1 $aLa Matematica per il 3+2,$x2038-5757 ;$v171 311 08$a3-031-85144-7 327 $a- Part I Classical ?-functions and applications -- 1. What is an ?-function? -- 2. The Prime Number Theorem -- 3. Review of Algebraic Number Theory -- 4. A Primer of Representation Theory -- 5. The ?-Function of a Complex Galois Representation -- 6. Dirichlet?s Theorem on Arithmetic Progressions -- 7. The Chebotarev Density Theorem -- Part II Prerequisites for Tate?s Thesis -- 8. The Haar Measure -- 9. Abstract Fourier Analysis -- 10. Review of Local Fields -- 11. Restricted Direct Products -- Part III Tate?s Thesis -- 12. The Local Theory -- 13. The Global Theory -- 14. Hecke ?-Functions -- 15. Recovering the Classical Theory -- 16. An Extended Example: the ?-Function of a CM Elliptic Curve. 330 $aThis book provides an accessible introduction to the theory of L-functions, emphasising their central role in number theory and their direct applications to key results. Designed to be elementary, it offers readers a clear pathway into the subject, starting from minimal background. It describes several important classes of L-functions ? Riemann and Dedekind zeta functions, Dirichlet L-functions, and Hecke L-functions (for characters with finite image) ? by showing how they are all special cases of the construction, due to Artin, of the L-function of a Galois representation. The analytic properties of abelian L-functions are presented in detail, including the full content of Tate's thesis, which establishes analytic continuation and functional equations via harmonic analysis. General Hecke L-functions are also discussed, using the modern perspective of idèles and adèles to connect their analytic theory with the representation-theoretic approach of Artin's L-functions. A distinguishing feature of this book is its accessibility: while largely avoiding arithmetic geometry, it provides introductions to both algebraic number theory and key aspects of representation theory. This approach ensures that the material is accessible to both beginning graduate students and advanced undergraduates. Applications play a central role throughout, highlighting how L-functions underpin significant results in number theory. The book provides complete proofs of the prime number theorem, Dirichlet's theorem on primes in arithmetic progressions, Chebotarev's density theorem, and the analytic class number formula, demonstrating the power of the theory in solving classical problems. It serves as an ideal introduction for advanced undergraduates and beginning graduate students and can also be a useful reference for preparing a course on the subject. 410 0$aLa Matematica per il 3+2,$x2038-5757 ;$v171 606 $aNumber theory 606 $aNumber Theory 615 0$aNumber theory. 615 14$aNumber Theory. 676 $a512.7 700 $aLombardo$b Davide$0613943 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910999786003321 996 $aL-Functions$94375488 997 $aUNINA