Combinatorics meets number theory in this stimulating stroll through the zetas. Includes well-chosen illustrations and exercises, both theoretical and computer-based.
Combinatorics meets number theory in this stimulating stroll through the zetas. Includes well-chosen illustrations and exercises, both theoretical and computer-based.
Audrey Terras is Professor of Mathematics at the University of California, San Diego.
Inhaltsangabe
List of illustrations Preface Part I. A Quick Look at Various Zeta Functions: 1. Riemann's zeta function and other zetas from number theory 2. Ihara's zeta function 3. Selberg's zeta function 4. Ruelle's zeta function 5. Chaos Part II. Ihara's Zeta Function and the Graph Theory Prime Number Theorem: 6. Ihara zeta function of a weighted graph 7. Regular graphs, location of poles of zeta, functional equations 8. Irregular graphs: what is the RH? 9. Discussion of regular Ramanujan graphs 10. The graph theory prime number theorem Part III. Edge and Path Zeta Functions: 11. The edge zeta function 12. Path zeta functions Part IV. Finite Unramified Galois Coverings of Connected Graphs: 13. Finite unramified coverings and Galois groups 14. Fundamental theorem of Galois theory 15. Behavior of primes in coverings 16. Frobenius automorphisms 17. How to construct intermediate coverings using the Frobenius automorphism 18. Artin L-functions 19. Edge Artin L-functions 20. Path Artin L-functions 21. Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function 22. The Chebotarev Density Theorem 23. Siegel poles Part V. Last Look at the Garden: 24. An application to error-correcting codes 25. Explicit formulas 26. Again chaos 27. Final research problems References Index.
List of illustrations Preface Part I. A Quick Look at Various Zeta Functions: 1. Riemann's zeta function and other zetas from number theory 2. Ihara's zeta function 3. Selberg's zeta function 4. Ruelle's zeta function 5. Chaos Part II. Ihara's Zeta Function and the Graph Theory Prime Number Theorem: 6. Ihara zeta function of a weighted graph 7. Regular graphs, location of poles of zeta, functional equations 8. Irregular graphs: what is the RH? 9. Discussion of regular Ramanujan graphs 10. The graph theory prime number theorem Part III. Edge and Path Zeta Functions: 11. The edge zeta function 12. Path zeta functions Part IV. Finite Unramified Galois Coverings of Connected Graphs: 13. Finite unramified coverings and Galois groups 14. Fundamental theorem of Galois theory 15. Behavior of primes in coverings 16. Frobenius automorphisms 17. How to construct intermediate coverings using the Frobenius automorphism 18. Artin L-functions 19. Edge Artin L-functions 20. Path Artin L-functions 21. Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function 22. The Chebotarev Density Theorem 23. Siegel poles Part V. Last Look at the Garden: 24. An application to error-correcting codes 25. Explicit formulas 26. Again chaos 27. Final research problems References Index.
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