Zeta Functions over Zeros of Zeta Functions

ISBN-13: 9783642052026 / Angielski / Miękka / 2009 / 163 str.

André Voros

Opis

The famous zeros of the Riemann zeta function and its generalizations (L-functions, Dedekind and Selberg zeta functions) are analyzed through several zeta functions built over those zeros. These 'second-generation' zeta functions have surprisingly many explicit, yet largely unnoticed properties, which are surveyed here in an accessible and synthetic manner, and then compiled in numerous tables. No previous book has addressed this neglected topic in analytic number theory. Concretely, this handbook will help anyone faced with symmetric sums over zeros like Riemann's. More generally, it aims at reviving the interest of number theorists and complex analysts toward those unfamiliar functions, on the 150th anniversary of Riemann's work.

Recenzje

From the reviews:

"The book is written in a pleasant style, with short chapters, each explaining a single topic. ... This book is a very timely synthesis of the results that have been scattered in the literature up to now. It gives a systematic overview and adds the discoveries of the author (with collaborators). ... this book will prove to be an important step in the further development of the subject of superzeta functions." (Machiel van Frankenhuijsen, Mathematical Reviews, Issue 2010 j)

"The book is very carefully written and presents an object of research which has been investigated for many decades but never treated in such a comprehensive manner. A valuable contribution to the field of analytic number theory." (Anton Deitmar, Zentralblatt MATH, Vol. 1206, 2011)

Spis treści

Infinite Products and Zeta-Regularization.- The Riemann Zeta Function #x03B6;(): a Primer.- Riemann Zeros and Factorizations of the Zeta Function.- Superzeta Functions: an Overview.- Explicit Formulae.- The Family of the First Kind {#x2112; ( | )}.- The Family of the Second Kind.- The Family of the Third Kind.- Extension to Other Zeta- and -Functions.- Application: an Asymptotic Criterion for the Riemann Hypothesis.