Kevin Broughan is Emeritus Professor in the Department of Mathematics and Statistics at the University of Waikato, New Zealand. In these two volumes he has used a unique combination of mathematical knowledge and skills. Following the publication of his Columbia University thesis, he worked on problems in topology before undertaking work on symbolic computation, leading to development of the software system SENAC. This led to a symbolic-numeric dynamical systems study of the zeta function, giving new insights into its behaviour, and was accompanied by publication of the software GL(n)pack as part of D. Goldfeld's book, Automorphic Forms and L-Functions for the Group GL(n,R). Professor Broughan has published widely on problems in prime number theory. His other achievements include co-establishing the New Zealand Mathematical Society, the School of Computing and Mathematical Sciences at the University of Waikato, and the basis for New Zealand's connection to the internet.
Inhaltsangabe
Volume 1. Analytic Equivalents: 1. Introduction 2. The Riemann Zeta function 3. Estimates 4. Classical equivalences 5. Euler's Totient function 6. A variety of abundant numbers 7. Robin's theorem 8. Numbers which do not satisfy Robin's inequality 9. Left, right and extremely abundant numbers 10. Other equivalents to the Riemann hypothesis Appendix A. Tables Appendix B. RHpack mini-manual Bibliography Index. Volume 2. Arithmetic Equivalents: 1. Introduction 2. Series equivalents 3. Banach and Hilbert space methods 4. The Riemann Xi function 5. The de Bruijn-Newman constant 6. Orthogonal polynomials 7. Cyclotomic polynomials 8. Integral equations 9. Weil's explicit formula, inequality and conjectures 10. Discrete measures 11. Hermitian forms 12. Dirichlet L-functions 13. Smooth numbers 14. Epilogue Appendix A. Convergence of series Appendix B. Complex function theory Appendix C. The Riemann-Stieltjes integral Appendix D. The Lebesgue integral on R Appendix E. Fourier transform Appendix F. The Laplace transform Appendix G. The Mellin transform Appendix H. The gamma function Appendix I. Riemann Zeta function Appendix J. Banach and Hilbert spaces Appendix K. Miscellaneous background results Appendix L. GRHpack mini-manual References Index.
Volume 1. Analytic Equivalents: 1. Introduction 2. The Riemann Zeta function 3. Estimates 4. Classical equivalences 5. Euler's Totient function 6. A variety of abundant numbers 7. Robin's theorem 8. Numbers which do not satisfy Robin's inequality 9. Left, right and extremely abundant numbers 10. Other equivalents to the Riemann hypothesis Appendix A. Tables Appendix B. RHpack mini-manual Bibliography Index. Volume 2. Arithmetic Equivalents: 1. Introduction 2. Series equivalents 3. Banach and Hilbert space methods 4. The Riemann Xi function 5. The de Bruijn-Newman constant 6. Orthogonal polynomials 7. Cyclotomic polynomials 8. Integral equations 9. Weil's explicit formula, inequality and conjectures 10. Discrete measures 11. Hermitian forms 12. Dirichlet L-functions 13. Smooth numbers 14. Epilogue Appendix A. Convergence of series Appendix B. Complex function theory Appendix C. The Riemann-Stieltjes integral Appendix D. The Lebesgue integral on R Appendix E. Fourier transform Appendix F. The Laplace transform Appendix G. The Mellin transform Appendix H. The gamma function Appendix I. Riemann Zeta function Appendix J. Banach and Hilbert spaces Appendix K. Miscellaneous background results Appendix L. GRHpack mini-manual References Index.
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