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The Theory of Hardy's Z-Function

Aleksandar Ivić
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Tracts in Mathematics 196
[Reviewed by
Mehdi Hassani
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The Riemann zeta function and its analytic properties have an essential role in analytic number theory, because there is a very close relationship between the locations of the non-real zeros of the Riemann zeta function and distribution of the prime numbers. There are other important functions related to the Riemann zeta function, including Hardy’s Z-function. This has the important property that the zeros of the Riemann zeta function on the “critical line” Re(s) = 1/2 are in one-to-one correspondence with the real zeros of Hardy’s Z-function. This makes Hardy’s Z-function a key tool in the study of the zeros of the Riemann zeta function on the critical line.

After his impressive work The Riemann Zeta-Function, Ivić uses Hardy’s Z-function as a pretext to write more deeply about the Riemann zeta function. The book starts with definitions of these functions and their basic properties but goes on to discuss some very technical concepts, properties and results.

The prerequisites to start reading this book are good standard courses in real and complex analysis. Each of 11 chapters of the book end with “Notes”, which are capsules of many references, comments, remarks, and historical notes. The book has no exercises, but if one follows the Notes in detail they would provide many very good exercises. Thus, the book could be used as a textbook at the graduate level. This book is full of fresh research directions, however, so it is really intended for researchers in the field. There is a rich bibliography.

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.

1. Definition of ζ(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 ζk(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