Trigonometric sums and their applications

This volume presents in a unified manner both classic as well as modern research results devoted to trigonometric sums. Such sums play an integral role in the formulation and understanding of a broad spectrum of problems which range over surprisingly many and different research areas. Fundamental and new developments are presented to discern solutions to problems across several scientific disciplines. Graduate students and researchers will find within this book numerous examples and a plethora of results related to trigonometric sums through pure and applied research along with open problems and new directions for future research. Preface......Page 5 Contents......Page 6 1 Introduction......Page 10 1.1 Nyman–Beurling Criterion for the Riemann Hypothesis......Page 11 1.2 The Cotangent Sum's Applications to Problems Related to the Riemann Hypothesis......Page 14 2 Central Properties of the Cotangent Sum c0......Page 17 2.1 Ellipse......Page 21 3 The Maximum of c0 in Rational Numbers in Short Intervals......Page 24 4 The Function g(x) and Moments of c0......Page 28 5 Dedekind Sums......Page 33 6 Sums Appearing in the Nyman-Beurling Criterion for the Riemann Hypothesis Containing the Möbius Function......Page 34 References......Page 36 Notation......Page 38 1 Ultraflat Sequences of Unimodular Polynomials......Page 40 2 More Recent Results on Ultraflat Sequences of UnimodularPolynomials......Page 45 3 Flatness of Conjugate-Reciprocal Unimodular Polynomials......Page 47 4 Average Lq Norm of Littlewood Polynomials on the Unit Circle......Page 49 5 Rudin-Shapiro Polynomials......Page 50 6 Mahler Measure and Moments of the Rudin-Shapiro Polynomials......Page 52 7 Lemmas for Theorem 6.1......Page 53 8 Saffari's Conjecture on the Shapiro Polynomials......Page 54 9 Consequences of Saffari's Conjecture......Page 55 10 Open Problems Related to the Rudin-Shapiro Polynomials......Page 58 11 On the Size of the Fekete Polynomials on the Unit Circle......Page 59 12 Unimodular Zeros of Self-Reciprocal Polynomials with Coefficients in a Finite Set......Page 62 13 Bourgain's L1 Problem and Related Results......Page 69 Reference......Page 73 1 Introduction and Statement of Main Results......Page 79 2 Lemmas......Page 82 3 Proof of Theorem 1......Page 85 4 Proof of Theorem 2......Page 86 5 Proof of Theorem 3......Page 89 7 Remarks......Page 90 References......Page 91 1 Introduction......Page 93 2 A Technical Lemma......Page 95 3 Main Results......Page 99 References......Page 103 1 Introduction......Page 105 2 Main Results......Page 108 3 Proof of Theorem 3......Page 110 4 The Discrete Versions of the Main Results......Page 115 5 Some Applications......Page 117 References......Page 125 1 Introduction......Page 126 2 Proof of Theorems 1.3 and 1.4......Page 136 References......Page 153 The Maximum of Cotangent Sums Related to the Nyman-Beurling Criterion for the Riemann Hypothesis......Page 155 1 Introduction......Page 156 2 Exponential Sums over Primes in Finite Fields......Page 159 3 Other Preliminary Lemmas......Page 160 4 Proof of Theorem 1.5......Page 163 References......Page 164 1 Introduction......Page 165 2 An Overview of the Results Related to Double-Sided Taylor's Approximations......Page 166 3.1 Generalization of Statement 1......Page 168 3.2 An Improvement of Statement 2......Page 170 References......Page 172 1 Introduction......Page 174 2 Auxiliary Lemmas......Page 176 3 Proof of the Main Theorem......Page 178 References......Page 187 1 Introduction and Preliminaries......Page 188 2 Lambert and Eisenstein Series......Page 191 2.1 Further Remarks and Observations for Eisenstein Series......Page 194 3 Dedekind Sums......Page 196 3.1 Some Others Formulas for the Dedekind Sums......Page 201 4 Hardy Sums......Page 203 5 Dedekind Type Daehee-Changhee (DC) Sums......Page 208 6 Trigonometric Representation of the DC-Sums......Page 210 7 DC-Sums Related to Special Functions......Page 213 8 Reciprocity Law......Page 215 9 Sums Obtained from Gauss-Chebyshev Quadratures......Page 221 10 Sums Obtained from Trigonometric Quadrature Rules......Page 228 References......Page 230 On a Half-Discrete Hilbert-Type Inequality in the Whole Plane with the Kernel of Hyperbolic Secant Function Related to the Hurwitz Zeta Function......Page 234 1 Introduction......Page 235 2 Weight Functions and Some Lemmas......Page 237 3 Main Results......Page 243 4 Operator Expressions......Page 250 5 Two Kinds of Equivalent Reverse Inequalities......Page 254 6 Conclusions......Page 262 References......Page 263 1 Introduction......Page 265 2 Definitions......Page 267 3 On the Multiplicative Energy of Sets with Small Wiener Norm......Page 269 4 On the Quantity M+......Page 273 References......Page 276 1 Introduction......Page 277 2 Best Orthogonal Trigonometric Approximations of the Classes Lψβ,p, 1