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 Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects of the theory, starting from first principles and probing the function's own challenging theory, with the famous and still unsolved "Riemann hypothesis" at its heart. The second edition has been revised to include descriptions of work done in the last forty years and is updated with many additional references; it will provide stimulating reading for postgraduates and workers in analytic number theory and classical analysis.

The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany

Zeta functions have been a powerful tool in mathematics over the last two centuries. This book considers a new class of non-commutative zeta functions which encode the structure of the subgroup lattice in infinite groups. The book explores the analytic behaviour of these functions together with an investigation of functional equations. Many important examples of zeta functions are calculated and recorded providing an important data base of explicit examples and methods for calculation.

Superb high-level study of one of the most influential classics in mathematics examines landmark 1859 publication entitled “On the Number of Primes Less Than a Given Magnitude,” and traces developments in theory inspired by it. Topics include Riemann's main formula, the prime number theorem, the Riemann-Siegel formula, large-scale computations, Fourier analysis, and other related topics. English translation of Riemann's original document appears in the Appendix.

In this book, I investigate (on a undergraduate level) the implication of 3D nontrivial zero solutions and its connection to the Montgomery Pair correlation conjecture. If their exist a 3D landscape to the nontrivial zeros (3D Riemann Hypothesis) then correspondingly their exist a 3D eigenvalue landscape. The arrangement of these 3D hypercomplex eigenvalue equivalent to 3D hypercomplex nontrivial zero solutions. What makes this so interesting is that this 3D eigenvalue landscape may be describing a new undiscovered 3D hypercomplex Quantum Mechanical landscape. I also explore other new discoveries on L-functions and the Prime Number Theorem.

This book provides a systematic account of several breakthroughs in the modern theory of zeta functions. It contains two different approaches to introduce and study genuine zeta functions for reductive groups (and their maximal parabolic subgroups) defined over number fields. Namely, the geometric one, built up from stability of principal lattices and an arithmetic cohomology theory, and the analytic one, from Langlands' theory of Eisenstein systems and some techniques used in trace formula, respectively. Apparently different, they are unified via a Lafforgue type relation between Arthur's analytic truncations and parabolic reductions of Harder–Narasimhan and Atiyah–Bott. Dominated by the stability condition and/or the Lie structures embedded in, these zeta functions have a standard form of the functional equation, admit much more refined symmetric structures, and most surprisingly, satisfy a weak Riemann hypothesis. In addition, two levels of the distributions for their zeros are exposed, i.e. a classical one giving the Dirac symbol, and a secondary one conjecturally related to GUE. This book is written not only for experts, but for graduate students as well. For example, it offers a summary of basic theories on Eisenstein series and stability of lattices and arithmetic principal torsors. The second part on rank two zeta functions can be used as an introduction course, containing a Siegel type treatment of cusps and fundamental domains, and an elementary approach to the trace formula involved. Being in the junctions of several branches and advanced topics of mathematics, these works are very complicated, the results are fundamental, and the theory exposes a fertile area for further research. Contents: Non-Abelian Zeta Functions Rank Two Zeta Functions Eisenstein Periods and Multiple L-Functions Zeta Functions for Reductive Groups Algebraic, Analytic Structures and Rieman Hypothesis Geometric Structures and Riemann Hypothesis Five Essays on Arithmetic Cohomology Readership: Graduate students and researchers in the theory of zeta functions. Keywords: Zeta Function;Riemann Hypothesis;Stability;Lattice;Fundamental Domain;Reductive Group;Root System;Eisenstein Series;Truncation;Arithmetic Principal Torsor;Adelic CohomologyReview: Key Features: Genuine zeta functions for reductive groups over number fields are introduced and studied systematically, based on (i) fine parabolic structures and Lie structures involved, (ii) a new stability theory for arithmetic principal torsors over number fields, and (iii) trace formula via a geometric understanding of Arthur's analytic truncations For the first time in history, we prove a weak Riemann hypothesis for zeta functions of reductive groups defined over number fields Not only the theory is explained, but the process of building the theory is elaborated in great detail

The Riemann zeta function was introduced by L. Euler (1737) in connection with questions about the distribution of prime numbers. Later, B. Riemann (1859) derived deeper results about the prime numbers by considering the zeta function in the complex variable. The famous Riemann Hypothesis, asserting that all of the non-trivial zeros of zeta are on a critical line in the complex plane, is one of the most important unsolved problems in modern mathematics. The present book consists of two parts. The first part covers classical material about the zeros of the Riemann zeta function with applications to the distribution of prime numbers, including those made by Riemann himself, F. Carlson, and Hardy-Littlewood. The second part gives a complete presentation of Levinson's method for zeros on the critical line, which allows one to prove, in particular, that more than one-third of non-trivial zeros of zeta are on the critical line. This approach and some results concerning integrals of Dirichlet polynomials are new. There are also technical lemmas which can be useful in a broader context.