Exploring the Riemann Zeta Function (eBook)

Preface by Freeman J. Dyson:Quasi-Crystals and the Riemann Hypothesis 5
References 7
Contents 8
An Introduction to Riemann's Life, His Mathematics, and His Work on the Zeta Function 10
1 Introduction 10
2 The Zeta Function 17
References 20
Ramanujan's Formula for ?(2n+1) 22
1 Introduction 22
2 Ramanujan's Unpublished Manuscript 24
3 An Alternative Formulation of (2) in Terms of Hyperbolic Cotangent Sums 28
4 Eisenstein Series 29
5 Eichler Integrals and Period Polynomials 31
6 Ramanujan's Formula for ?(2n+1) and Euler's Formula for ?(2n) are Consequences of the Same General Theorem 34
7 The Associated Polynomials and Their Roots 37
8 Further History of Proofs of (2) 40
References 40
Towards a Fractal Cohomology: Spectra of Polya–Hilbert Operators, Regularized Determinants and Riemann Zeros 44
1 Introduction 45
1.1 The Weil Conjectures 45
1.2 Polya–Hilbert Operators and a Cohomology Theory in Characteristic Zero 47
1.3 Fractal Cohomology 48
2 Background 49
3 Derivative Operator on Weighted Bergman Spaces 52
4 The Construction 58
4.1 Refining the Operator of a Meromorphic Function 61
5 Applications of the Construction 63
5.1 Rational Functions 63
5.2 Zeta Functions of Curves over Finite Fields 64
5.3 The Gamma Function 65
5.4 The Riemann Zeta Function 66
5.5 Hadamard's Factorization Theorem of Entire Functions 69
6 Conclusion 71
References 72
The Temptation of the Exceptional Characters 75
1 Introduction 75
2 The Class Number Formula 75
3 The Size of L(1, ?D) 76
4 The Zero-Free Region 77
5 The Class Number 77
6 Primes in Arithmetic Progressions 79
7 Twin Primes 80
8 More Primes in Arithmetic Progressions 81
9 Primes in Short Intervals 81
10 Prime Values of Polynomials 82
11 Squares Plus Primes 82
12 Some Ideas About the Proofs 83
13 The Exceptional Conductor Influencing Itself 84
14 Influence on the Nearby Conductors 86
References 88
Arthur's Truncated Eisenstein Series for SL(2, Z) and the Riemann Zeta Function: A Survey 90
1 Introduction 90
2 Arthur's Truncated Eisenstein Series 93
3 Fourier Expansion of Arthur's Truncated Eisenstein Series 95
4 Maass–Selberg Relation for SL(2, Z) 98
5 A Zero Free Region for the Riemann Zeta Function 100
References 103
On a Cubic Moment of Hardy's Function with a Shift 105
1 Introduction 105
2 Statement of Results 108
3 Proof of the Theorems 109
References 117
Some Analogues of Pair Correlation of Zeta Zeros 119
1 Introduction 119
2 A Sketch of Montgomery's Derivation 123
3 A Modified Approach 124
4 Pair Correlation of Zeta Zeros 125
5 Preliminaries Concerning the Relative Maxima of |?(12+it)| 127
6 Preliminaries Concerning the Iterated Convolutions of the von Mangoldt Function 132
7 Correlation of Zeta Zeros with the Relative Maxima of |?(12+it)| 155
8 Pair Correlation of Zeros of Z1(s) 165
9 Correlation of the Zeros of Two Dirichlet L-Functions 179
10 Further Problems to Study 183
References 184
Bagchi's Theorem for Families of Automorphic Forms 186
1 Introduction 186
Acknowledgements 188
Notation 188
2 Equidistribution and Universality for Modular Forms in the Level Aspect 188
3 Proof of Theorem 2 191
4 Proof of Theorem 3 200
5 Generalizations 202
References 203
The Liouville Function and the Riemann Hypothesis 205
1 Introduction 205
2 Generalizing the Problems of Pólya and Turán 209
3 The Möbius Function and the Riemann Hypothesis 213
4 Weak Independence 215
5 Lattices and Improved Oscillation Bounds 217
6 Computations 219
7 Improvements 222
References 224
Explorations in the Theory of Partition Zeta Functions 226
1 The Setting: Visions of Euler 226
1.1 Partition-Theoretic Zeta Functions 228
1.2 Evaluations 232
1.2.1 Zeta Functions for Partitions with Parts Restricted by Congruence Conditions 232
1.2.2 Connections to Ordinary Riemann Zeta Values 233
1.2.3 Zeta Functions for Partitions of Fixed Length 234
1.3 Analytic Continuation and p-Adic Continuity 235
1.4 Connections to Multiple Zeta Values 236
1.5 Machinery 238
1.5.1 Useful Formulas 238
1.5.2 Proofs of Theorems 1 and 2, and Their Corollaries 240
1.5.3 Proof of Theorem 3 and Its Corollaries 242
1.5.4 Proofs of Results Concerning Multiple Zeta Values 245
1.6 Some Further Thoughts 246
2 Zeta Polynomials 247
2.1 Partitions and Modular Forms 247
2.2 Manin's Zeta Polynomials 250
2.3 Tools Needed for the Proof of Manin's Conjecture 258
2.3.1 Work of Rodriguez-Villegas 258
2.3.2 Period Polynomials at 1 259
2.3.3 Ingredients for Theorem 10 259
2.4 Sketch of Proofs for Theorems on Zeta Polynomials and the Riemann Hypothesis for Period Polynomials 260
2.5 Examples 262
2.5.1 Period Polynomials in Weight 4 262
2.5.2 Zeta Function for the Modular Discriminant 263
2.5.3 Ehrhart Polynomials and Newforms of Weight 6 264
2.6 Concluding Remark 265
References 266
Reading Riemann 268
1 Introduction 268
2 Prime Number Theory Before Riemann's Paper 272
3 In the Intermezzo 274
4 Riemann's Mode of Writing 277
5 The Paper on Prime Numbers 279
6 The Zeta Function and Bessel Functions 283
7 Changes in Our Understanding of Riemann 286
Reference 288
A Taniyama Product for the Riemann Zeta Function 289
1 Introduction 289
2 Taniyama's Identity 291
3 The Modification 292
4 The Analogue of Mertens' Theorem 293
5 The Special Value 296
6 A Question 297
References 300