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This book is an outgrowth of the conference “Regulators IV: An International Conference on Arithmetic L-functions and Differential Geometric Methods” that was held in Paris in May 2016. Gathering contributions by leading experts in the field ranging from original surveys to pure research articles, this volume provides comprehensive coverage of the front most developments in the field of regulator maps. Key topics covered are: • Additive polylogarithms • Analytic torsions • Chabauty-Kim theory • Local Grothendieck-Riemann-Roch theorems • Periods • Syntomic regulator The book contains contributions by M. Asakura, J. Balakrishnan, A. Besser, A. Best, F. Bianchi, O. Gregory, A. Langer, B. Lawrence, X. Ma, S. Müller, N. Otsubo, J. Raimbault, W. Raskin, D. Rössler, S. Shen, N. Triantafi llou, S. Ünver and J. Vonk.

Number theory. --- Differential geometry. --- Algebraic geometry. --- Number Theory. --- Differential Geometry. --- Algebraic Geometry. --- Algebraic geometry --- Geometry --- Differential geometry --- Number study --- Numbers, Theory of --- Algebra --- L-functions --- Functions, L --- -Number theory --- Funcions L --- Geometria diferencial --- Geometria --- Càlcul de tensors --- Connexions (Matemàtica) --- Coordenades --- Corbes --- Cossos convexos --- Dominis convexos --- Espais de curvatura constant --- Espais simètrics --- Estructures hermitianes --- Formes diferencials --- G-estructures --- Geodèsiques (Matemàtica) --- Geometria de Riemann --- Geometria diferencial global --- Geometria integral --- Geometria simplèctica --- Hiperespai --- Subvarietats (Matemàtica) --- Topologia diferencial --- Varietats (Matemàtica) --- Varietats de Kähler --- L-Funcions --- Teoria de nombres

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Teoria de nombres --- Teoria dels nombres --- Àlgebra --- Anàlisi diofàntica --- Arrels de la unitat --- Congruències i residus --- Conjectura de Catalan --- Darrer teorema de Fermat --- Formes automorfes --- Formes quadràtiques --- Fórmula de traça de Selberg --- Funcions aritmètiques --- Funcions L --- Funcions modulars --- Funcions recursives --- Funcions zeta --- Geometria algebraica aritmètica --- Geometria de nombres --- Grups modulars --- Lleis de reciprocitat --- Nombres de Fermat --- Nombres ordinals --- Nombres p-àdics --- Nombres transfinits --- Numeració --- Particions (Matemàtica) --- Quadrats màgics --- Sedàs (Matemàtica) --- Teorema de Fermat --- Teorema de Gödel --- Teoria algebraica de nombres --- Teoria de Galois --- Cossos algebraics --- Number theory. --- Number study --- Numbers, Theory of --- Algebra

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This textbook offers a unique exploration of analytic number theory that is focused on explicit and realistic numerical bounds. By giving precise proofs in simplified settings, the author strategically builds practical tools and insights for exploring the behavior of arithmetical functions. An active learning style is encouraged across nearly three hundred exercises, making this an indispensable resource for both students and instructors. Designed to allow readers several different pathways to progress from basic notions to active areas of research, the book begins with a study of arithmetic functions and notions of arithmetical interest. From here, several guided “walks” invite readers to continue, offering explorations along three broad themes: the convolution method, the Levin–Faĭnleĭb theorem, and the Mellin transform. Having followed any one of the walks, readers will arrive at “higher ground”, where they will find opportunities for extensions and applications, such as the Selberg formula, Exponential sums with arithmetical coefficients, and the Large Sieve Inequality. Methodology is emphasized throughout, with frequent opportunities to explore numerically using computer algebra packages Pari/GP and Sage. Excursions in Multiplicative Number Theory is ideal for graduate students and upper-level undergraduate students who are familiar with the fundamentals of analytic number theory. It will also appeal to researchers in mathematics and engineering interested in experimental techniques in this active area.

Teoria de nombres --- Multiplicació --- Multiplication. --- Arithmetic --- Ready-reckoners --- Aritmètica --- Teoria dels nombres --- Àlgebra --- Anàlisi diofàntica --- Arrels de la unitat --- Congruències i residus --- Conjectura de Catalan --- Darrer teorema de Fermat --- Formes automorfes --- Formes quadràtiques --- Fórmula de traça de Selberg --- Funcions aritmètiques --- Funcions L --- Funcions modulars --- Funcions recursives --- Funcions zeta --- Geometria algebraica aritmètica --- Geometria de nombres --- Grups modulars --- Lleis de reciprocitat --- Nombres de Fermat --- Nombres ordinals --- Nombres p-àdics --- Nombres transfinits --- Numeració --- Particions (Matemàtica) --- Quadrats màgics --- Sedàs (Matemàtica) --- Teorema de Fermat --- Teorema de Gödel --- Teoria algebraica de nombres --- Teoria de Galois --- Cossos algebraics --- Number theory. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra

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Number theory. --- Geometria algebraica aritmètica --- Teoria de nombres --- Number study --- Numbers, Theory of --- Algebra --- Teoria dels nombres --- Àlgebra --- Anàlisi diofàntica --- Arrels de la unitat --- Congruències i residus --- Conjectura de Catalan --- Darrer teorema de Fermat --- Formes automorfes --- Formes quadràtiques --- Fórmula de traça de Selberg --- Funcions aritmètiques --- Funcions L --- Funcions modulars --- Funcions recursives --- Funcions zeta --- Geometria de nombres --- Grups modulars --- Lleis de reciprocitat --- Nombres de Fermat --- Nombres ordinals --- Nombres p-àdics --- Nombres transfinits --- Numeració --- Particions (Matemàtica) --- Quadrats màgics --- Sedàs (Matemàtica) --- Teorema de Fermat --- Teorema de Gödel --- Teoria algebraica de nombres --- Teoria de Galois --- Cossos algebraics --- Geometria algèbrica aritmètica --- Geometria diofàntica --- Geometria algebraica --- Punts racionals (Geometria) --- Varietats de Shimura

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Ever since the analogy between number fields and function fields was discovered, it has been a source of inspiration for new ideas, and a long history has not in any way detracted from the appeal of the subject. As a deeper understanding of this analogy could have tremendous consequences, the search for a unified approach has become a sort of Holy Grail. The arrival of Arakelov's new geometry that tries to put the archimedean places on a par with the finite ones gave a new impetus and led to spectacular success in Faltings' hands. There are numerous further examples where ideas or techniques from the more geometrically-oriented world of function fields have led to new insights in the more arithmetically-oriented world of number fields, or vice versa. These invited articles by leading researchers in the field explore various aspects of the parallel worlds of function fields and number fields. Topics range from Arakelov geometry, the search for a theory of varieties over the field with one element, via Eisenstein series to Drinfeld modules, and t-motives. This volume is aimed at a wide audience of graduate students, mathematicians, and researchers interested in geometry and arithmetic and their connections. Contributors: G. Böckle; T. van den Bogaart; H. Brenner; F. Breuer; K. Conrad; A. Deitmar; C. Deninger; B. Edixhoven; G. Faltings; U. Hartl; R. de Jong; K. Köhler; U. Kühn; J.C. Lagarias; V. Maillot; R. Pink; D. Roessler; and A. Werner.

Geometry, Algebraic. --- Mathematical physics. --- Number theory. --- Number study --- Numbers, Theory of --- Physical mathematics --- Physics --- Algebraic geometry --- Mathematics --- Mathematics. --- Algebraic geometry. --- Physics. --- Algebraic Geometry. --- Number Theory. --- Mathematical Methods in Physics. --- Algebra --- Geometry --- Geometry, algebraic. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Teoria de nombres --- Cossos algebraics --- Camp (Matemàtica) --- Camps (Matemàtica) --- Camps algebraics --- Camps algèbrics --- Cos (Matemàtica) --- Cossos (Matemàtica) --- Cossos algèbrics --- Nombres algebraics --- Nombres algèbrics --- Teoria de camps --- Teoria de camps (Àlgebra) --- Teoria de camps (Matemàtica) --- Teoria de cossos --- Teoria de cossos (Àlgebra) --- Teoria de cossos (Matemàtica) --- Teoria algebraica de nombres --- Àlgebra diferencial --- Anells de divisió --- Camps finits (Àlgebra) --- Cossos topològics --- Extensions de cossos (Matemàtica) --- Ideals (Àlgebra) --- Teoria dels nombres --- Àlgebra --- Anàlisi diofàntica --- Arrels de la unitat --- Congruències i residus --- Conjectura de Catalan --- Darrer teorema de Fermat --- Formes automorfes --- Formes quadràtiques --- Fórmula de traça de Selberg --- Funcions aritmètiques --- Funcions L --- Funcions modulars --- Funcions recursives --- Funcions zeta --- Geometria algebraica aritmètica --- Geometria de nombres --- Grups modulars --- Lleis de reciprocitat --- Nombres de Fermat --- Nombres ordinals --- Nombres p-àdics --- Nombres transfinits --- Numeració --- Particions (Matemàtica) --- Quadrats màgics --- Sedàs (Matemàtica) --- Teorema de Fermat --- Teorema de Gödel --- Teoria de Galois