Arithmetic Tales

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.

• algebraic number fields
• asymptotics for arithmetical functions
• elementary and multiplicative number theory
• exponential sums estimates
• zeta functions

From the reviews:

“The book under review should succeed very well as a source from which to learn a lot of very beautiful number theory in an accessible way … . This book is clearly a true labor of love and Bordellès has produced an important text and an elegant scholarly work.” (Michael Berg, The Mathematical Association of America, August, 2012)

“The book is essentially self-contained and, in such a manner, a suitable primer for upper-level undergraduates. But also masters-level students and more advanced graduates will find a wealth of fundamental and fascinating arithmetic topics in this panoramic textbook. … It is very gratifying to have an English version of this excellent, textbook of number theory by O. Bordellès, and that in the present, significantly extended and improved form. No doubt, this book will find many interested readers within the international mathematical community.” (Werner Kleinert, Zentralblatt MATH, Vol. 1244, 2012)

“Number theory constitutes a super-subject, a virtual commonwealth of distinctive and semiautonomous disciplines all connected together by a volume of cross-border commerce sufficient to preclude any effective decoupling. … Modern developments that have a reasonably elementary character … appear in a logical way alongside vintage topics. … Replete with novel observations, clever exercises (solved!), and many nice details. Summing Up: Highly recommended. Lower-division undergraduates and above.” (D. V. Feldman, Choice, Vol. 50 (7), March, 2013)

“The author has written an introduction to number theory which starts with elementary questions and quickly advances to more difficult subjects. … At the end of each chapter there is a section containing rich information about recent developments and a list of exercises. The book contains a lot of useful information, and may be of interest also for specialists.” (W. Narkiewicz, Mathematical Reviews, January,2013)

• Aiguilhe, France

  Olivier Bordellès

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