Number Theory in Memory of Eduard Wirsing

Preface
Contents
Life and Work of Eduard Wirsing
	1 A Brief Biography
	2 The Early Years 1950–1957: Additive Number Theory
	3 The Postdoc Period 1958–1969: Primes and Approximations
	4 Thirty Years as Professor 1969–1999: From Logarithms to Partitions
	5 Retirement and No End: Lattice Points, Rigidity, and More
	6 Isolated, Unpublished and Late Results
	References
Remembering Eduard Wirsing
	References
Personal Memories
On the Infimum of the Absolute Value of Successive Derivatives of a Real Function Defined on a Bounded Interval
	1 Introduction
	2 First Observations
		2.1 Homogeneity
		2.2 An Extremal Problem
	3 The Relevance of Monic Polynomials
	4 Extremal Polynomials
	5 Results on C(n,p)
		5.1 The Case n=1
		5.2 The Case p=∞
		5.3 The Case p=2
		5.4 The Case p=1
		5.5 Bounds for C(n,p)
		5.6 An Open Question
	References
Friable Averages of Oscillating Arithmetic Functions
	1 Introduction and Statements of Results
	2 Applications
		2.1 Weighted Averages
		2.2 Truncated Multiplicative Functions
	3 Solutions to Delay-Differential Equations
		3.1 The Function hκ
		3.2 The Functions φκ and ψκ
		3.3 Link with Positive Convolution Powers of the Dickman Function
	4 Proof of Theorem 1.1
		4.1 Auxiliary Estimates
		4.2 Completion of the Proof
	5 Proof of Theorem 1.2
		5.1 The Case κN*
		5.2 The Case κN*
	6 Proof of Theorem 2.1
	7 Proof of Corollary 1.4
	8 Proof of Theorem 2.2
	References
Ein quaternäres Waring-Goldbach-Problem
	1 Die Frage
	2 Ein Mittelwertsatz
	3 Die Kreismethode
	4 Abschied von der Krone
	5 Der Rückschnitt
	Literatur
Coprimality of Consecutive Elements in a Piatetski-Shapiro Sequence
	1 Introduction
	2 Proof of Theorem 1.1
		2.1 A Little Lemma
		2.2 Reduction of Theorem 1.1
		2.3 Proof of Theorem 1.1
	3 Proof of Theorem 1.2
		3.1 Reduction
		3.2 Proof of Theorem 1.2
	References
Wirsing\'s Elementary Proofs of the Prime Number Theorem with Remainder Terms
	1 Historical Background
	2 The Selberg Formula Mechanism
	3 Error Estimates, I
	4 Error Estimates, II
	References
Diophantine Analysis Around [1,2,3,…]
	1 Introduction of the Zopf-Number
	2 Rational Convergents of the Zopf-Number
	3 Quadratic Convergents of the Zopf-Number
		3.1 An Application to the Zopf-Number
		3.2 Expressing Quadratic Convergents by Numerators and Denominators of Rational Convergents
		3.3 On the Approximation of the Zopf-Number by Quadratic Convergents
		3.4 Proof of Lemma 3.4
		3.5 Proof of Lemma 3.7
	4 Error Sums of the Zopf-Number
		4.1 Preliminaries
		4.2 Main Results
		4.3 Proofs
	5 Concluding Comments
	References
On a Smoothed Average of the Number of Goldbach Representations
	1 Introduction and Statement of Results
	2 Lemmas
	3 Proof of Theorem 1.1
	4 Transition from Fq(N) to Gq(N)
	References
Estimates for k-Dimensional Spherical Summations of Arithmetic Functions of the GCD and LCM
	1 Introduction
	2 Preliminaries
		2.1 Known Estimates for the Number of Lattice Points in k-Dimensional Spheres
		2.2 Sums of Functions of the GCD and LCM over the Integers, Respectively Natural Numbers
		2.3 Some Lemmas
	3 Main Results
		3.1 Spherical Summations of Arbitrary Functions
		3.2 Estimates for Functions of the GCD
		3.3 Estimates for Functions of the LCM
	4 Proofs
		4.1 Proofs of the Lemmas
		4.2 Proof of the Generalization of Wintner\'s Theorem
		4.3 Proofs of the Results for Functions of the GCD
		4.4 Proofs of the Results for Functions of the LCM
	References
The Rational Points Close to a Space Curve
	References
Twists by Dirichlet Characters and Polynomial Euler Productsof L-Functions, II
	1 Introduction
	2 Definitions
	3 Lemmas
	4 Proof of Theorem 1.1
	5 Proof of Theorem 1.2
	References
Solving the Iterative Differential Equation -γg\' = g-1
	1 Introduction
	2 The Operator T
	3 Intersection Behaviour and Main Results
	References
Irrationality of Zeros of the Digamma Function
	1 Introduction
	2 Preliminary Results on the Digamma Function
	3 The Digamma Function at Rational Arguments
	4 Zeros of ψ(x)+γ and Proofs of Theorems 1.1 and 1.2
	5 Proof of Theorem 1.3
	6 Concluding Remarks
	References
Generalizations of Menon\'s Arithmetic Identity
	1 Menon\'s Identity
	2 The Eckford Cohen Totient Function φ(k)(m)
	3 Pillai\'s Sum Functions P(k)(m)
	4 Proof of Theorem 1.4
	References
On a Conjecture of Descartes
	1 Introduction
	2 Notation: The Role of the Explicit Formula
	3 Contribution of the Minor Arcs
	4 Proof of Theorem 1.1
	References
On the Greatest Common Divisor of a Number and Its Sumof Divisors, II
	1 Introduction
		1.1 Perfect Numbers
		1.2 The Distribution of gcd(n,σ(n))
		1.3 A Word on Notation
	2 The Frequency of n with gcd(n,σ(n))=m
	3 Proof of Theorem 1.1
		3.1 Preliminaries Concerning Smooth Numbers
		3.2 Lower Bounds in Theorems 1.1 and 1.2
		3.3 Upper Bound in Theorem 1.2
		3.4 Upper Bound in Theorem 1.1
	References
Permutations with Arithmetic Constraints
	1 Introduction
	2 An Upper Bound for #Slcm(n)
	3 Lower Bounds
	4 Comparing #Sdiv(n) and #Slcm(n)
	5 An Upper Bound for Anti-coprime Permutations
	Dedication and Acknowledgments
	References
Large Subsums of the Möbius Function
	1 Introduction
	2 Proof of Theorem 1.1
	3 Prolegomena to Theorem 1.2
	4 Proof of Theorem 1.2
	References
The a-Points of the Riemann Zeta-Function and the FunctionalEquation
	1 Motivation and Statement of the Main Results
	2 Preliminaries About a-Points and Gonek\'s Lemma
	3 Proof of the Special Case a=0
	4 Proof of Theorem 1.1 in Case a≠0
	5 Proof of Theorem 1.2 and a Research Question
	References
Braided Gibonacci Sequences on Residue Classes
	1 Introduction and Results
	2 Auxiliary Results
	3 Braided Gibonacci Sequences
	4 Restriction on a Residue Class and Proof of Theorem 1.1
	References