Zeta Functions over Zeros of Zeta Functions

In this text, 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. 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. 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)…