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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.
This book is designed both for FPGA users interested in developing new, specific components - generally for reducing execution times –and IP core designers interested in extending their catalog of specific components. The main focus is circuit synthesis and the discussion shows, for example, how a given algorithm executing some complex function can be translated to a synthesizable circuit description, as well as which are the best choices the designer can make to reduce the circuit cost, latency, or power consumption. This is not a book on algorithms. It is a book that shows how to translate efficiently an algorithm to a circuit, using techniques such as parallelism, pipeline, loop unrolling, and others. Numerous examples of FPGA implementation are described throughout this book and the circuits are modeled in VHDL. Complete and synthesizable source files are available for download.
"This monograph is devoted to arithmetic functions, an area of number theory. Arithmetic functions are very important in many parts of theoretical and applied sciences, and many mathematicians have devoted great interest in this field. One of the interesting features of this book is the introduction and study of certain new arithmetic functions that have been considered by the authors separately or together, and their importance is shown in many connections with the classical arithmetic functions or in their applications to other problems"--
This volume focuses on the classical theory of number-theoretic functions emphasizing algebraic and multiplicative techniques. It contains many structure theorems basic to the study of arithmetic functions, including several previously unpublished proofs. The author is head of the Dept. of Mathemati
An extensive summary of mathematical functions that occur in physical and engineering problems
This volume focuses on the classical theory of number-theoretic functions emphasizing algebraic and multiplicative techniques. It contains many structure theorems basic to the study of arithmetic functions, including several previously unpublished proofs. The author is head of the Dept. of Mathemati
Aimed at presenting nontechnical explanations, all the essays in this collection of papers from the 1989 LMS Durham Symposium on L-functions are the contributions of renowned algebraic number theory specialists.
"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
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.
The theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their number-theoretical aspects. After two chapters geared toward elementary levels, there follows a detailed treatment of the theory of Hecke operators, which associate zeta functions to modular forms. At a more advanced level, complex multiplication of elliptic curves and abelian varieties is discussed. The main question is the construction of abelian extensions of certain algebraic number fields, which is traditionally called "Hilbert's twelfth problem." Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by modular functions, which supplies an indispensable background for the recent proof of Fermat's last theorem by Wiles.