The theory of Hardy's Z-function
QA241 .I83 2013
- Unknown QA241 .I83 2013
- Includes bibliographical references (pages 225-238) and indexes.
- 1. Definition of zeta(s), Z(t) and basic notions-- 2. The zeros on the critical line-- 3. The Selberg class of L-functions-- 4. The approximate functional equations for zetak(s)-- 5. The derivatives of Z(t)-- 6. Gram points-- 7. The moments of Hardy's function-- 8. The primitive of Hardy's function-- 9. The Mellin transforms of powers of Z(t)-- 10. Further results on Mk(s)$ and Zk(s)-- 11. On some problems involving Hardy's function and zeta moments-- References-- Index.
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- Publisher's Summary
- Hardy's Z-function, related to the Riemann zeta-function zeta(s), was originally utilised by G. H. Hardy to show that zeta(s) has infinitely many zeros of the form 1/2+it. It is now amongst the most important functions of analytic number theory, and the Riemann hypothesis, that all complex zeros lie on the line 1/2+it, is perhaps one of the best known and most important open problems in mathematics. Today Hardy's function has many applications; among others it is used for extensive calculations regarding the zeros of zeta(s). This comprehensive account covers many aspects of Z(t), including the distribution of its zeros, Gram points, moments and Mellin transforms. It features an extensive bibliography and end-of-chapter notes containing comments, remarks and references. The book also provides many open problems to stimulate readers interested in further research.
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- Supplemental links
- Cover image
- Number theory.
- Publication date
- Copyright date
- Aleksandar Ivić, Univerzitet u Beogradu, Serbia.
- Cambridge tracts in mathematics ; 196
- 9781107028838 (hardback)
- 1107028833 (hardback)