Zeta Functions over Zeros of Zeta Functions: (8 Lecture Notes of the Unione Matematica Italiana)

In the Riemann zeta function ?(s), the non-real zeros or Riemann zeros, denoted ?, play an essential role mainly in number theory, and thereby g- erate considerable interest. However, they are very elusive objects. Thus, no individual zero has an analytically known location; and the Riemann - pothesis, which states that all those zeros should lie on the critical line, i.e., 1 haverealpart ,haschallengedmathematicianssince1859(exactly150years 2 ago). For analogous symmetric sets of numbers{v}, such as the roots of a k polynomial,the eigenvalues of a ?nite or in?nite matrix,etc., it is well known that symmetric functions of the{v} tend to have more accessible properties k than the individual elements v . And, we ?nd the largest wealth of explicit k properties to occur in the (generalized) zeta functions of the generic form ?s Zeta(s,a)= (v +a) k k (with the extra option of replacing v here by selected functions f(v )). k k Not surprisingly, then, zeta functions over the Riemann zeros have been considered, some as early as 1917.What is surprising is how small the lite- ture on those zeta functions has remained overall.We were able to spot them in barely a dozen research articles over the whole twentieth century and in none ofthebooks featuring the Riemannzeta function. So the domainexists, but it has remained largely con?dential and sporadically covered, in spite of a recent surge of interest. Could it then be that those zeta functions have few or uninteresting pr- erties?Inactualfact,theirstudyyieldsanabundanceofquiteexplicitresults.

Table of Contents:
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.

Review :
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)