Trigonometric Sums and Their Applications

Preface
Contents
On a Category of Cotangent Sums Related to the Nyman-Beurling Criterion for the Riemann Hypothesis
	1 Introduction
		1.1 Nyman–Beurling Criterion for the Riemann Hypothesis
		1.2 The Cotangent Sum\'s Applications to Problems Related to the Riemann Hypothesis
	2 Central Properties of the Cotangent Sum c0
		2.1 Ellipse
	3 The Maximum of c0 in Rational Numbers in Short Intervals
	4 The Function g(x) and Moments of c0
	5 Dedekind Sums
	6 Sums Appearing in the Nyman-Beurling Criterion for the Riemann Hypothesis Containing the Möbius Function
	References
Recent Progress in the Study of Polynomials with Constrained Coefficients
	Notation
	1 Ultraflat Sequences of Unimodular Polynomials
	2 More Recent Results on Ultraflat Sequences of UnimodularPolynomials
	3 Flatness of Conjugate-Reciprocal Unimodular Polynomials
	4 Average Lq Norm of Littlewood Polynomials on the Unit Circle
	5 Rudin-Shapiro Polynomials
	6 Mahler Measure and Moments of the Rudin-Shapiro Polynomials
	7 Lemmas for Theorem 6.1
	8 Saffari\'s Conjecture on the Shapiro Polynomials
	9 Consequences of Saffari\'s Conjecture
	10 Open Problems Related to the Rudin-Shapiro Polynomials
	11 On the Size of the Fekete Polynomials on the Unit Circle
	12 Unimodular Zeros of Self-Reciprocal Polynomials with Coefficients in a Finite Set
	13 Bourgain\'s L1 Problem and Related Results
	Reference
Classes of Nonnegative Sine Polynomials
	1 Introduction and Statement of Main Results
	2 Lemmas
	3 Proof of Theorem 1
	4 Proof of Theorem 2
	5 Proof of Theorem 3
	6 Proof of Theorem 4
	7 Remarks
	References
Inequalities for Weighted Trigonometric Sums
	1 Introduction
	2 A Technical Lemma
	3 Main Results
	References
Norm Inequalities for Generalized Laplace Transforms
	1 Introduction
	2 Main Results
	3 Proof of Theorem 3
	4 The Discrete Versions of the Main Results
	5 Some Applications
	References
On Marcinkiewicz-Zygmund Inequalities at Hermite Zeros and Their Airy Function Cousins
	1 Introduction
	2 Proof of Theorems 1.3 and 1.4
	References
The Maximum of Cotangent Sums Related to the Nyman-Beurling Criterion for the Riemann Hypothesis
	1 Introduction
	2 Exponential Sums over Primes in Finite Fields
	3 Other Preliminary Lemmas
	4 Proof of Theorem 1.5
	References
Double-Sided Taylor\'s Approximations and Their Applications in Theory of Trigonometric Inequalities
	1 Introduction
	2 An Overview of the Results Related to Double-Sided Taylor\'s Approximations
	3 Main Results
		3.1 Generalization of Statement 1
		3.2 An Improvement of Statement 2
	4 Conclusion
	References
The Second Moment of the First Derivative of Hardy\'s Z-Function
	1 Introduction
	2 Auxiliary Lemmas
	3 Proof of the Main Theorem
	References
Dedekind and Hardy Type Sums and Trigonometric Sums Induced by Quadrature Formulas
	1 Introduction and Preliminaries
	2 Lambert and Eisenstein Series
		2.1 Further Remarks and Observations for Eisenstein Series
	3 Dedekind Sums
		3.1 Some Others Formulas for the Dedekind Sums
	4 Hardy Sums
	5 Dedekind Type Daehee-Changhee (DC) Sums
	6 Trigonometric Representation of the DC-Sums
	7 DC-Sums Related to Special Functions
	8 Reciprocity Law
	9 Sums Obtained from Gauss-Chebyshev Quadratures
	10 Sums Obtained from Trigonometric Quadrature Rules
	References
On a Half-Discrete Hilbert-Type Inequality in the Whole Plane with the Kernel of Hyperbolic Secant Function Related to the Hurwitz Zeta Function
	1 Introduction
	2 Weight Functions and Some Lemmas
	3  Main Results
	4 Operator Expressions
	5 Two Kinds of Equivalent Reverse Inequalities
	6 Conclusions
	References
A Remark on Sets with Small Wiener Norm
	1 Introduction
	2 Definitions
	3 On the Multiplicative Energy of Sets with Small Wiener Norm
	4 On the Quantity M+
	References
Order Estimates of Best Orthogonal Trigonometric Approximations of Classes of Infinitely Differentiable Functions
	1 Introduction
	2 Best Orthogonal Trigonometric Approximations of the Classes Lψβ,p, 1