Equivalents of the Riemann Hypothesis: Volume 3, Further Steps towards Resolving the Riemann Hypothesis (Encyclopedia of Mathematics and its Applications, Series Number 187)

Cover Half-title Page Series Page Title Page Imprints Page Dedication Epigraph Contents Preface 1 Nicolas’ π(x) < li(θ(x)) Equivalence 1.1 Introduction 1.2 Estimating the Logarithmic Integral 1.3 The Function A[sub(1)](x) 1.4 The Functions B(x) and A[sub(2)](x) 1.5 Asymptotic and Explicit Bounds for the Function A(x) 1.6 A Big Omega Theorem of Robin 1.7 End Note 2 Nicolas’ Number of Divisors Function Equivalence 2.1 Introduction 2.2 Results Derived from Chapter 1 2.3 Superior Highly Composite Numbers 2.4 Preliminary Lemmas 2.5 Technical Lemmas 2.6 Proof of Nicolas’ Explicit Inequality Assuming RH 2.7 Benefit and Convexity 2.8 The Main Theorem 2.9 End Note 3 An Aspect of the Zeta Function Zero Gap Estimates 3.1 Introduction 3.2 Preliminary Results 3.3 Gonek’s Sum of Powers to Zeta Zeros 3.4 Main Theorem 4 The Rogers–Tao Equivalence 4.1 Introduction 4.2 Definitions and Overview 4.3 Preliminary Results 4.4 Counting the Zeros of H[sub(t)](z) 4.5 A Lower Bound on Gaps Between Zeros 4.6 Asymptotics for the Integral of the Energy 4.7 Evolution of the Adjusted Hamiltonian 4.8 Estimates for the Hamiltonian and Energy 4.9 The Fundamental Lemma and Main Theorem 5 The Dirichlet Series of Dobner 5.1 Introduction 5.2 Preliminary Lemmas 5.3 Fundamental Lemma and Theorem 5.4 Main Result 5.5 Dobner’s Theorem for an Extended Selberg Class 5.6 End Note 6 An Upper Bound for the de Bruijn–Newman Constant 6.1 Introduction 6.2 Imported Results 6.3 Definitions 6.4 Notation 6.5 Zero Dynamics 6.6 Basic Estimates 6.7 Fundamental Lemma and Theorem 6.8 Essential Estimates 6.9 The Unbounded Region 6.10 The Barrier Region 6.11 The Bounded Region 6.12 Criteria for an Upper Bound 6.13 The Main Theorem 6.14 End Note 7 The Pólya–Jensen Equivalence 7.1 Introduction 7.2 Jensen Polynomials and the Class LP 7.3 The Pólya–Jensen Equivalence for RH 7.4 The Work of Csordas et al. 7.5 The Result of Chasse 8 Ono et al. and Jensen Polynomials 8.1 Introduction 8.2 Hermite Polynomials 8.3 The Linear Growth Condition 8.4 Asymptotics for the ξ(s) MacLaurin Coefficients 8.5 The Linear Growth Condition for the Coefficients 8.6 The Second Article of Ono et al. 8.7 The Results’s Reception 8.8 David Farmer’s Response 8.9 Conrey and Gosh’s Example 8.10 End Note 9 Gonek–Bagchi Universality and Bagchi’s Equivalence 9.1 Introduction 9.2 Notations 9.3 Gonek’s Proof Overview 9.4 Gonek’s First, Fundamental Lemma 9.5 Technical Lemmas 9.6 Gonek’s Second Fundamental Lemma 9.7 Gonek’s Universal Property Theorem 9.8 Corollaries to Gonek’s Theorem 9.9 Bagchi’s Lemma and RH Equivalence Overview 9.10 Bagchi’s RH Equivalence 9.11 End Note 10 A Selection of Undecidable Propositions 10.1 Introduction 10.2 Poonen’s List and Other Undecidable Examples 10.3 Semi-Thue Systems 10.4 Tag Systems 10.5 Hilbert’s 10th Problem is Undecidable 10.6 Some Undecidable Consequences of DPRM 10.7 Laczkovich’s Undecidable Example 10.8 Congruential Functions 10.9 Conway’s Unpredictable Iterations 11 Equivalences and Decidability for Riemann’s Zeta 11.1 Introduction 11.2 RH and the Arithmetic Hierarchy 11.3 Matiyasevich’s Polynomial RH Equivalence 11.4 Matiyasevich’s Integer Equivalence 11.5 Ramsey Theory 11.6 The Paris–Harrington and Sine Principles 11.7 Paris–Harrington Theorem Proof 11.8 Bovykin–Weiermann Preliminary Results 11.9 The Bovykin–Weiermann Theorems 11.10 Applications of Recursive Function Theory to RH 11.11 Epilogue Appendix A Imports for Gonek’s Theorems Appendix B Imports for Nicolas’ Theorems B.1 Theorems B.2 Superior Highly Composite Numbers Appendix C Hyperbolic Polynomials C.1 Introduction C.2 Results of Schur, Marlo, and Obrechkoff C.3 Borcea–Branden’s and Chasse’s Theorems Appendix D Absolute Continuity Appendix E Montel’s and Hurwitz’s Theorems Appendix F Markov’s and Gronwall’s Inequalities Appendix G Characterizing Riemann’s Zeta Function Appendix H Bohr’s Theorem Appendix I Zeta and L-Functions I.1 Introduction I.2 The Selberg Class I.3 Properties of the Selberg Class I.4 Selberg’s Conjectures I.5 Consequences of the Selberg Definitions and Conjectures I.7 Dokchitser L-functions I.8 Iwaniec–Kowalski Class I.9 Consequences of These Conditions I.10 Examples Appendix J de Reyna’s Expansion for the Hardy Contour J.1 Riemann–Siegel Formula J.2 Riemann’s Integral for Zeta J.3 Arias De Reyna’s Expansion Appendix K Stirling’s Approximation for the Gamma Function K.1 Introduction K.2 Polymath15’s Estimate Appendix L Propositional Calculus P[sub(0)] L.1 Introduction L.2 A Brief Account of the Beginnings of Mathematical Logic L.3 Propositional Calculus L.4 The System P[sub(0)] Appendix M First Order Predicate Calculus P[sub(1)] M.1 Introduction M.2 First, Order Mathematical Theories M.3 Examples of First-Order Theories M.4 Models and Truth in an Interpretation M.5 Logical Axioms and Rules of Inference for P[sub(1)] M.6 Theorems for P[sub(1)] M.7 Decidability in P[sub(1)] M.8 Some Mathematical Applications M.9 Models and the Compactness Theorem M.10 Gödel’s Incompleteness Theorem M.11 Arithmetic M.11.1 Peano Arithmetic PA M.11.2 Compact Arithmetic CA M.11.3 Presburger Arithmetic PR M.11.4 Skolem Arithmetic SK M.11.5 Robinson Arithmetic Q M.11.6 Takeuti’s Conservative Extension PAT M.12 Arithmetical Hierarchy Appendix N Recursive Functions N.1 Introduction N.2 Partial Recursive and Primitive Recursive Functions N.3 Decidable Predicates N.4 Recursively Enumerable Subsets N.5 Enumeration and Rice’s Theorem N.6 Algorithms and Machines N.7 Turing Machines N.8 Minsky Machines Appendix O Ordinal Numbers and Analysis O.1 Introduction O.2 Ordinal Numbers O.3 Primitive Recursive Arithmetic PRA O.4 Gentzen’s Consistency of Arithmetic O.5 The Ordinal Strength of Theories O.6 Paris–Harrington Theorem Proof using Ordinals O.7 End Note References Index