Download or read book Introduction to Arithmetical Functions written by Paul J. McCarthy and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 373 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of arithmetical functions has always been one of the more active parts of the theory of numbers. The large number of papers in the bibliography, most of which were written in the last forty years, attests to its popularity. Most textbooks on the theory of numbers contain some information on arithmetical functions, usually results which are classical. My purpose is to carry the reader beyond the point at which the textbooks abandon the subject. In each chapter there are some results which can be described as contemporary, and in some chapters this is true of almost all the material. This is an introduction to the subject, not a treatise. It should not be expected that it covers every topic in the theory of arithmetical functions. The bibliography is a list of papers related to the topics that are covered, and it is at least a good approximation to a complete list within the limits I have set for myself. In the case of some of the topics omitted from or slighted in the book, I cite expository papers on those topics.
Download or read book Arithmetical Functions written by Komaravolu Chandrasekharan and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: The plan of this book had its inception in a course of lectures on arithmetical functions given by me in the summer of 1964 at the Forschungsinstitut fUr Mathematik of the Swiss Federal Institute of Technology, Zurich, at the invitation of Professor Beno Eckmann. My Introduction to Analytic Number Theory has appeared in the meanwhile, and this book may be looked upon as a sequel. It presupposes only a modicum of acquaintance with analysis and number theory. The arithmetical functions considered here are those associated with the distribution of prime numbers, as well as the partition function and the divisor function. Some of the problems posed by their asymptotic behaviour form the theme. They afford a glimpse of the variety of analytical methods used in the theory, and of the variety of problems that await solution. I owe a debt of gratitude to Professor Carl Ludwig Siegel, who has read the book in manuscript and given me the benefit of his criticism. I have improved the text in several places in response to his comments. I must thank Professor Raghavan Narasimhan for many stimulating discussions, and Mr. Henri Joris for the valuable assistance he has given me in checking the manuscript and correcting the proofs. K. Chandrasekharan July 1970 Contents Chapter I The prime number theorem and Selberg's method § 1. Selberg's fonnula . . . . . . 1 § 2. A variant of Selberg's formula 6 12 § 3. Wirsing's inequality . . . . . 17 § 4. The prime number theorem. .
Download or read book Arithmetical Functions written by Wolfgang Schwarz and published by Cambridge University Press. This book was released on 1994-03-10 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: Characterizes certain multiplicative and additive arithmetical functions by combining methods from number theory with simple ideas from functional and harmonic analysis.
Download or read book Introduction to Arithmetical Functions written by Paul Joseph McCarthy and published by . This book was released on 1986-01-01 with total page 365 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Arithmetic Functions and Integer Products written by P.D.T.A. Elliott and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 469 pages. Available in PDF, EPUB and Kindle. Book excerpt: Every positive integer m has a product representation of the form where v, k and the ni are positive integers, and each Ei = ± I. A value can be given for v which is uniform in the m. A representation can be computed so that no ni exceeds a certain fixed power of 2m, and the number k of terms needed does not exceed a fixed power of log 2m. Consider next the collection of finite probability spaces whose associated measures assume only rational values. Let hex) be a real-valued function which measures the information in an event, depending only upon the probability x with which that event occurs. Assuming hex) to be non negative, and to satisfy certain standard properties, it must have the form -A(x log x + (I - x) 10g(I -x». Except for a renormalization this is the well-known function of Shannon. What do these results have in common? They both apply the theory of arithmetic functions. The two widest classes of arithmetic functions are the real-valued additive and the complex-valued multiplicative functions. Beginning in the thirties of this century, the work of Erdos, Kac, Kubilius, Turan and others gave a discipline to the study of the general value distribution of arithmetic func tions by the introduction of ideas, methods and results from the theory of Probability. I gave an account of the resulting extensive and still developing branch of Number Theory in volumes 239/240 of this series, under the title Probabilistic Number Theory.
Download or read book Various Arithmetic Functions and their Applications written by Octavian Cira and published by Infinite Study. This book was released on 2016 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes squares cubes factorials, almost primes, mobile periodicals, functions, tables, prime square factorial bases, generalized factorials, generalized palindromes, so on, have been extracted from the Archives of American Mathematics (University of Texas at Austin) and Arizona State University (Tempe): "The Florentin Smarandache papers" special collections, and Arhivele Statului (Filiala Vâlcea & Filiala Dolj, Romania). This book was born from the collaboration of the two authors, which started in 2013. The first common work was the volume "Solving Diophantine Equations", published in 2014. The contribution of the authors can be summarized as follows: Florentin Smarandache came with his extraordinary ability to propose new areas of study in number theory, and Octavian Cira - with his algorithmic thinking and knowledge of Mathcad.
Download or read book An Introduction to the Theory of Numbers written by Leo Moser and published by The Trillia Group. This book was released on 2004 with total page 95 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties, greatest common divisor, etc.), is an expanded version of a series of lectures for graduate students on elementary number theory. Topics include: Compositions and Partitions; Arithmetic Functions; Distribution of Primes; Irrational Numbers; Congruences; Diophantine Equations; Combinatorial Number Theory; and Geometry of Numbers. Three sections of problems (which include exercises as well as unsolved problems) complete the text."--Publisher's description
Download or read book Introduction to Analytic Number Theory written by Tom M. Apostol and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory. For this reason, the book starts with the most elementary properties of the natural integers. Nevertheless, the text succeeds in presenting an enormous amount of material in little more than 300 pages."-—MATHEMATICAL REVIEWS
Download or read book Higher Arithmetic written by Harold M. Edwards and published by American Mathematical Soc.. This book was released on 2008 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: Among the topics featured in this textbook are: congruences; the fundamental theorem of arithmetic; exponentiation and orders; primality testing; the RSA cipher system; polynomials; modules of hypernumbers; signatures of equivalence classes; and the theory of binary quadratic forms. The book contains exercises with answers.
Download or read book Arithmetical Functions written by Wolfgang Schwarz and published by . This book was released on 1994 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Arithmetic Tales written by Olivier Bordellès and published by Springer Science & Business Media. This book was released on 2012-05-31 with total page 569 pages. Available in PDF, EPUB and Kindle. Book excerpt: Number theory was once famously labeled the queen of mathematics by Gauss. The multiplicative structure of the integers in particular deals with many fascinating problems some of which are easy to understand but very difficult to solve. In the past, a variety of very different techniques has been applied to further its understanding. Classical methods in analytic theory such as Mertens’ theorem and Chebyshev’s inequalities and the celebrated Prime Number Theorem give estimates for the distribution of prime numbers. Later on, multiplicative structure of integers leads to multiplicative arithmetical functions for which there are many important examples in number theory. Their theory involves the Dirichlet convolution product which arises with the inclusion of several summation techniques and a survey of classical results such as Hall and Tenenbaum’s theorem and the Möbius Inversion Formula. Another topic is the counting integer points close to smooth curves and its relation to the distribution of squarefree numbers, which is rarely covered in existing texts. Final chapters focus on exponential sums and algebraic number fields. A number of exercises at varying levels are also included. Topics in Multiplicative Number Theory introduces offers a comprehensive introduction into these topics with an emphasis on analytic number theory. Since it requires very little technical expertise it will appeal to a wide target group including upper level undergraduates, doctoral and masters level students.
Download or read book Analytic Number Theory For Undergraduates written by Heng Huat Chan and published by World Scientific Publishing Company. This book was released on 2009-04-21 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is written for undergraduates who wish to learn some basic results in analytic number theory. It covers topics such as Bertrand's Postulate, the Prime Number Theorem and Dirichlet's Theorem of primes in arithmetic progression.The materials in this book are based on A Hildebrand's 1991 lectures delivered at the University of Illinois at Urbana-Champaign and the author's course conducted at the National University of Singapore from 2001 to 2008.
Download or read book Function Field Arithmetic written by Dinesh S. Thakur and published by World Scientific. This book was released on 2004 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an exposition of function field arithmetic with emphasis on recent developments concerning Drinfeld modules, the arithmetic of special values of transcendental functions (such as zeta and gamma functions and their interpolations), diophantine approximation and related interesting open problems. While it covers many topics treated in 'Basic Structures of Function Field Arithmetic' by David Goss, it complements that book with the inclusion of recent developments as well as the treatment of new topics such as diophantine approximation, hypergeometric functions, modular forms, transcendence, automata and solitons. There is also new work on multizeta values and log-algebraicity. The author has included numerous worked-out examples. Many open problems, which can serve as good thesis problems, are discussed.
Download or read book written by and published by . This book was released on 2007 with total page 435 pages. Available in PDF, EPUB and Kindle. Book excerpt: 本书内容包括素数、无理数、同余、费马定理、连分数、不定方程、二次域、算术函数、分化等。
Download or read book The Book of Prime Number Records written by Paulo Ribenboim and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 492 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text originated as a lecture delivered November 20, 1984, at Queen's University, in the undergraduate colloquium series established to honour Professors A. J. Coleman and H. W. Ellis and to acknowledge their long-lasting interest in the quality of teaching undergraduate students. In another colloquium lecture, my colleague Morris Orzech, who had consulted the latest edition of the Guinness Book oj Records, reminded me very gently that the most "innumerate" people of the world are of a certain tribe in Mato Grosso, Brazil. They do not even have a word to express the number "two" or the concept of plurality. "Yes Morris, I'm from Brazil, but my book will contain numbers different from 'one.' " He added that the most boring 800-page book is by two Japanese mathematicians (whom I'll not name), and consists of about 16 million digits of the number 11. "I assure you Morris, that in spite of the beauty of the apparent randomness of the decimal digits of 11, I'll be sure that my text will also include some words." Acknowledgment. The manuscript of this book was prepared on the word processor by Linda Nuttall. I wish to express my appreciation for the great care, speed, and competence of her work. Paulo Ribenboim CONTENTS Preface vii Guiding the Reader xiii Index of Notations xv Introduction Chapter 1. How Many Prime Numbers Are There? 3 I. Euclid's Proof 3 II.
Download or read book Abstract analytic number theory written by Knopfmacher and published by Newnes. This book was released on 2009-02-04 with total page 333 pages. Available in PDF, EPUB and Kindle. Book excerpt: North-Holland Mathematical Library, Volume 12: Abstract Analytic Number Theory focuses on the approaches, methodologies, and principles of the abstract analytic number theory. The publication first deals with arithmetical semigroups, arithmetical functions, and enumeration problems. Discussions focus on special functions and additive arithmetical semigroups, enumeration and zeta functions in special cases, infinite sums and products, double series and products, integral domains and arithmetical semigroups, and categories satisfying theorems of the Krull-Schmidt type. The text then ponders on semigroups satisfying Axiom A, asymptotic enumeration and "statistical" properties of arithmetical functions, and abstract prime number theorem. Topics include asymptotic properties of prime-divisor functions, maximum and minimum orders of magnitude of certain functions, asymptotic enumeration in certain categories, distribution functions of prime-independent functions, and approximate average values of special arithmetical functions. The manuscript takes a look at arithmetical formations, additive arithmetical semigroups, and Fourier analysis of arithmetical functions, including Fourier theory of almost even functions, additive abstract prime number theorem, asymptotic average values and densities, and average values of arithmetical functions over a class. The book is a vital reference for researchers interested in the abstract analytic number theory.
Download or read book Introduction to Modern Number Theory written by Yu. I. Manin and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 519 pages. Available in PDF, EPUB and Kindle. Book excerpt: This edition has been called ‘startlingly up-to-date’, and in this corrected second printing you can be sure that it’s even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories.
