دانلود کتاب Trigonometric sums and their applications

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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 3 Best Orthogonal Trigonometric Approximations of the Classes Lψβ,1 in the Metric of Space L∞......Page 286
4 Best Orthogonal Trigonometric Approximations of the Classes Lψβ,1 in the Metric of Spaces Ls, 1 References......Page 290
1 Introduction......Page 292
2 An Example and Two Lemmas......Page 295
3 Main Results......Page 299
4 Some Corollaries......Page 303
References......Page 307
Index......Page 309