To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure email@example.com
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
This text has grown out of a mini-course held at the Arctic Number Theory School, University of Helsinki, May 18–25, 2011. The centralt opic is Hardy's function Z(t), of great importance in the theory of the Riemann zeta-function ζ(s). It is named after Godfrey Harold (“G. H.”) Hardy FRS (1877–1947), who was a prominent English mathematician, well-known for his achievements in number theory and mathematical analysis. Sometimes by Hardy function(s) one denotes the element(s) of Hardys paces Hp, which are certain spaces of holomorphic functions on the unit disk or the upper half-plane. In this text, however, Hardy's function Z(t) will always denote the function defined by (0) below. It was chosen as the object of study because of its significance in the theory of ζ(s) and because, initially, considerable material could be presented on the blackboard within the f ramework of six lectures. Some results, like Theorem 6.7 and the bounds in (4.25) and (4.26) are new, improving on older ones. It is “Hardy's function” which is the thread that holds this work together. I have thought it is appropriate for a monograph because the topic is not as vast as the topic of the Riemann zeta-function itself.
Hardy's Z-function, related to the Riemann zeta-function ζ(s), was originally utilised by G. H. Hardy to show that ζ(s) has infinitely many zeros of the form ½+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 ½+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 ζ(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.