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E-raamat: Number Theory - Diophantine Problems, Uniform Distribution and Applications: Festschrift in Honour of Robert F. Tichy's 60th Birthday

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  • Ilmumisaeg: 26-May-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319553573
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Number Theory - Diophantine Problems, Uniform Distribution and Applications: Festschrift in Honour of Robert F. Tichy's 60th BirthdaySuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 26-May-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319553573
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This volume is dedicated to Robert F. Tichy on the occasion of his 60th birthday. Presenting 22 research and survey papers written by leading experts in their respective fields, it focuses on areas that align with Tichy"s research interests and which he significantly shaped, including Diophantine problems, asymptotic counting, uniform distribution and discrepancy of sequences (in theory and application), dynamical systems, prime numbers, and actuarial mathematics. Offering valuable insights into recent developments in these areas, the book will be of interest to researchers and graduate students engaged in number theory and its applications.

Preface.- Contents.- List of Contributors.- On Nearly Linear Recurrence Sequences, Shigeki Akiyama, Jan-Hendrik Evertse and Attila Pethö.- Risk Theory with Affine Dividend Payment Strategies, Hansjörg Albrecher and Arian Cani.- A Discrepancy Problem: Balancing Infinite Dimensional Vectors, József Beck.- Squares with Three Nonzero Digits, Michael A. Bennett and Adrian-Maria Scheerer.- On the density of coprime tuples of the form (n; bf1(n)c; : : : ; bfk(n)c), where f1; : : : ; fk are functions from a Hardy field, Vitaly Bergelson and Florian Karl Richter.- On the uniform theory of lacunary series, István Berkes.- Diversity in Parametric Families of Number Fields, Yuri Bilu and Florian Luca.- Local Oscillations in Moderately Dense Sequences of Primes, Jörg Brüdern and Christian Elsholtz.- Sums of the digits in bases 2 and 3, Jean-Marc Deshouillers, Laurent Habsieger, Shanta Laishram and Bernard Landreau.- On the Discrepancy of Halton-Kronecker Sequences, Michael Drmota, Roswitha Hof

er and Gerhard Larcher.- More on Diophantine Sextuples, Andrej Dujella and Matija Kazalicki.- Effective Results for Discrimant Equations Over Finitely Generated Integral Domains, Jan-Hendrik Evertse and Kálmán Györy.- Quasi-equivalence of Heights and Runge"s Theorem, Philipp Habegger.- On the Monoid Generated by a Lucas Sequence, Clemens Heuberger and Stephan Wagner.- Measures of pseudorandomness: Arithmetic Autocorrelation and correlation Measure, Richard Hofer, László Mérai and Arne Winterhof.- On Multiplicative Independent Bases for Canonical Number Systems in Cyclotomic Number Fields, Manfred G. Madritsch, Paul Surer and Volker Ziegler.- Refined Estimates for Exponential Sums and a Problem Concerning the Product of Three L-Series, Werner Georg Nowak.- Orbits of Algebraic Dynamical Systems in Subgroups and Subfields, Alina Ostafe and Igor E. Shparlinski.- Patterns of Primes in Arithmetic Progressions, János Pintz.- On Simple Linear Recurrences, Andrzej Schinzel.- Equivalence of

the Logarithmically Averaged Chowla and Sarnak Conjectures, Terence Tao.- Discrepancy Bounds for -Adic Halton Sequences, Jörg M. Thuswaldner.
1 Introduction and Historical Remarks
1(6)
2 Basic Number Theory
7(52)
2.1 The Ring of Integers
7(3)
2.2 Divisibility, Primes, and Composites
10(6)
2.3 The Fundamental Theorem of Arithmetic
16(6)
2.4 Congruences and Modular Arithmetic
22(17)
2.4.1 Basic Theory of Congruences
22(1)
2.4.2 The Ring of Integers Mod N
23(4)
2.4.3 Units and the Euler Phi Function
27(5)
2.4.4 Fermat's Little Theorem and the Order of an Element
32(4)
2.4.5 On Cyclic Groups
36(3)
2.5 The Solution of Polynomial Congruences Modulo m
39(9)
2.5.1 Linear Congruences and the Chinese Remainder Theorem
39(6)
2.5.2 Higher Degree Congruences
45(3)
2.6 Quadratic Reciprocity
48(7)
2.7 Exercises
55(4)
3 The Infinitude of Primes
59(84)
3.1 The Infinitude of Primes
59(33)
3.1.1 Some Direct Proofs and Variations
59(3)
3.1.2 Some Analytic Proofs and Variations
62(4)
3.1.3 The Fermat and Mersenne Numbers
66(5)
3.1.4 The Fibonacci Numbers and the Golden Section
71(13)
3.1.5 Some Simple Cases of Dirichlet's Theorem
84(5)
3.1.6 A Topological Proof and a Proof Using Codes
89(3)
3.2 Sums of Squares
92(20)
3.2.1 Pythagorean Triples
93(3)
3.2.2 Fermat's Two-Square Theorem
96(4)
3.2.3 The Modular Group
100(7)
3.2.4 Lagrange's Four Square Theorem
107(3)
3.2.5 The Infinitude of Primes Through Continued Fractions
110(2)
3.3 Dirichlet's Theorem
112(19)
3.4 Twin Prime Conjecture and Related Ideas
131(1)
3.5 Primes Between x and 2x
132(1)
3.6 Arithmetic Functions and the Mobius Inversion Formula
133(5)
3.7 Exercises
138(5)
4 The Density of Primes
143(76)
4.1 The Prime Number Theorem---Estimates and History
143(4)
4.2 Chebyshev's Estimate and Some Consequences
147(12)
4.3 Equivalent Formulations of the Prime Number Theorem
159(10)
4.4 The Riemann Zeta Function and the Riemann Hypothesis
169(17)
4.4.1 The Real Zeta Function of Euler
170(5)
4.4.2 Analytic Functions and Analytic Continuation
175(4)
4.4.3 The Riemann Zeta Function
179(7)
4.5 The Prime Number Theorem
186(7)
4.6 The Elementary Proof
193(5)
4.7 Multiple Zeta Values
198(8)
4.8 Some Extensions and Comments
206(7)
4.9 Exercises
213(6)
5 Primality Testing---An Overview
219(66)
5.1 Primality Testing and Factorization
219(1)
5.2 Sieving Methods
220(16)
5.2.1 Brun's Sieve and Bran's Theorem
226(10)
5.3 Primality Testing and Prime Records
236(27)
5.3.1 Pseudo-Primes and Probabilistic Testing
241(8)
5.3.2 The Lucas--Lehmer Test and Prime Records
249(6)
5.3.3 Some Additional Primality Tests
255(2)
5.3.4 Elliptic Curve Methods
257(6)
5.4 Cryptography and Primes
263(7)
5.4.1 Some Number Theoretic Cryptosystems
267(3)
5.5 Public Key Cryptography and the RSA Algorithm
270(3)
5.6 Elliptic Curve Cryptography
273(3)
5.7 The AKS Algorithm
276(6)
5.8 Exercises
282(3)
6 Primes and Algebraic Number Theory
285(86)
6.1 Algebraic Number Theory
285(2)
6.2 Unique Factorization Domains
287(21)
6.2.1 Euclidean Domains and the Gaussian Integers
293(8)
6.2.2 Principal Ideal Domains
301(3)
6.2.3 Prime and Maximal Ideals
304(4)
6.3 Algebraic Number Fields
308(21)
6.3.1 Algebraic Extensions of Q
316(3)
6.3.2 Algebraic and Transcendental Numbers
319(2)
6.3.3 Symmetric Polynomials
321(4)
6.3.4 Discriminant and Norm
325(4)
6.4 Algebraic Integers
329(19)
6.4.1 The Ring of Algebraic Integers
331(2)
6.4.2 Integral Bases
333(2)
6.4.3 Quadratic Fields and Quadratic Integers
335(4)
6.4.4 The Transcendence of e and π
339(3)
6.4.5 The Geometry of Numbers---Minkowski Theory
342(3)
6.4.6 Dirichlet's Unit Theorem
345(3)
6.5 The Theory of Ideals
348(18)
6.5.1 Unique Factorization of Ideals
350(7)
6.5.2 An Application of Unique Factorization
357(2)
6.5.3 The Ideal Class Group
359(2)
6.5.4 Norms of Ideals
361(3)
6.5.5 Class Number
364(2)
6.6 Exercises
366(5)
7 The Fields Qp of p-Adic Numbers: Hensel's Lemma
371(34)
7.1 The p-Adic Fields and p-Adic Expansions
371(2)
7.2 The Construction of the Real Numbers
373(8)
7.2.1 The Completeness of Real Numbers
373(3)
7.2.2 The Construction of R
376(5)
7.2.3 The Characterization of R
381(1)
7.3 Normed Fields and Cauchy Completions
381(1)
7.4 The p-Adic Fields
382(5)
7.4.1 The p-Adic Norm
385(2)
7.5 The Construction of Qp
387(7)
7.5.1 p-Adic Arithmetic and p-Adic Expansions
387(7)
7.6 The p-Adic Integers
394(4)
7.6.1 Principal Ideals and Unique Factorization
396(1)
7.6.2 The Completeness of Zp
397(1)
7.7 Ostrowski's Theorem
398(1)
7.8 Hensel's Lemma and Applications
398(5)
7.8.1 The Non-isomorphism of the p-Adic Fields
402(1)
7.9 Exercises
403(2)
Bibliography 405(4)
Index 409
Christian Elsholtz is Associate Professor at the Graz University of Technology, Institute of Analysis and Number Theory. Peter Grabner is Professor at the Graz University of Technology, Institute of Analysis and Number Theory.
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