Summing It Up: From One Plus One to Modern Number Theory by Avner Ash (PDF)

Description

The power and properties of numbers, from basic addition and sums of squares to cutting-edge theoryWe use addition on a daily basis―yet how many of us stop to truly consider the enormous and remarkable ramifications of this mathematical activity? Summing It Up uses addition as a springboard to present a fascinating and accessible look at numbers and number theory, and how we apply beautiful numerical properties to answer math problems. Mathematicians Avner Ash and Robert Gross explore addition’s most basic characteristics as well as the addition of squares and other powers before moving onward to infinite series, modular forms, and issues at the forefront of current mathematical research.Ash and Gross tailor their succinct and engaging investigations for math enthusiasts of all backgrounds. Employing college algebra, the first part of the book examines such questions as, can all positive numbers be written as a sum of four perfect squares? The second section of the book incorporates calculus and examines infinite series―long sums that can only be defined by the concept of limit, as in the example of 1+1/2+1/4+. . .=? With the help of some group theory and geometry, the third section ties together the first two parts of the book through a discussion of modular forms―the analytic functions on the upper half-plane of the complex numbers that have growth and transformation properties. Ash and Gross show how modular forms are indispensable in modern number theory, for example in the proof of Fermat’s Last Theorem.Appropriate for numbers novices as well as college math majors, Summing It Up delves into mathematics that will enlighten anyone fascinated by numbers.

User’s Reviews

Editorial Reviews: Review “Offers a clear and beautiful progression from addition to modern number theory.” ― Math-Blog”The authors did a remarkable job in making some aspects of modern number theory very accessible to readers with only a minimal knowledge of mathematics, say a student who had a first calculus course. However, also mathematicians who do not have number theory as their main focus will enjoy this book.”—Adhemar Bultheel, European Mathematical Society”Ash and Gross do a masterful job of leading students from finite sums to modular forms and to the forefront of modern number theory. . . . This is an excellent piece of mathematical writing.” ― Choice”[A]n accessible and fun introduction to modular forms. . . . [Summing It Up] is engaging and conversational, without losing accuracy or essential rigor.”—Dominic Lanphier, American Mathematical Monthly”Summing It Up uses addition as a springboard to present a fascinating and accessible look at numbers and number theory, and how we apply beautiful numerical properties to answer math problems. . . . Ash and Gross tailor their succinct and engaging investigations for math enthusiasts of all backgrounds.” ― L’Enseignement Mathematique”Readers of the books [Fearless Symmetry and Elliptic Tales] will not need this review to be talked into buying ‘Summing it Up by the same authors, Avner Ash and Robert Gross. . . . Like the other volumes it takes the readers on a guided tour through parts of modern number theory by starting with very elementary notions.”—Franz Lemmermeyer, Zentralblatt MATH”With their customary charm and characteristic panache, the authors manage mostly to be clear and precise without being pedantic. . . . There is no doubt that a conscientious [reader] will gain real understanding of the many topics being discussed.”—Peter Shiu, Mathematical Gazette”Engaging and conversational, without losing accuracy or essential rigor.”—Dominic Lanphier, The American Mathematical Monthly Review “In Summing It Up, Ash and Gross, with their customary charm and clarity, build an impressive tower from the most basic mathematical story imaginable. They start by adding up finite sequences and whole numbers before journeying to the upper reaches of contemporary developments in number theory.”―Jordan Ellenberg, author of How Not to Be Wrong: The Power of Mathematical Thinking”With joy and reflection,Summing It Up encourages readers to work alongside the authors to engage in simple computations with profound implications. It progresses elegantly to an insightful discussion of the ideas behind the classical theory of modular forms. A wonderful book.”―Barry Mazur, Harvard University”This book takes general readers on a journey from simple addition to the modern theory of modular forms. Plentiful and accurate references are available for interested readers who want to look up further information.”―Xiaoheng Wang, Princeton University”In Summing It Up, Ash and Gross are erudite and meticulous―they write with care and precision. Their book adopts the voice of a wise elder telling math stories to precocious children. It will interest readers of all ages who seek a window into current mathematics.”―Kenneth Ribet, University of California, Berkeley”Summing It Up ambitiously presents concepts of number theory, from the elementary to the advanced, for readers with only a knowledge of high school math and some calculus. With a crisp yet conversational style and excellent examples, Ash and Gross explain a great amount of interesting and important math.”―James Pommersheim, coauthor of Number Theory: A Lively Introduction with Proofs, Applications, and Stories”A well-motivated and fun introduction to some very deep topics in mathematics, Summing It Up takes readers on a mathematics tour starting with very classical material and building towards some amazing recent work, with many highlights along the way. It can be read by those with many different levels of mathematical background. I thoroughly enjoyed it.”―Nathan Kaplan, University of California, Irvine From the Back Cover “In Summing It Up, Ash and Gross, with their customary charm and clarity, build an impressive tower from the most basic mathematical story imaginable. They start by adding up finite sequences and whole numbers before journeying to the upper reaches of contemporary developments in number theory.”–Jordan Ellenberg, author of How Not to Be Wrong: The Power of Mathematical Thinking”With joy and reflection, Summing It Up encourages readers to work alongside the authors to engage in simple computations with profound implications. It progresses elegantly to an insightful discussion of the ideas behind the classical theory of modular forms. A wonderful book.”–Barry Mazur, Harvard University”This book takes general readers on a journey from simple addition to the modern theory of modular forms. Plentiful and accurate references are available for interested readers who want to look up further information.”–Xiaoheng Wang, Princeton University”In Summing It Up, Ash and Gross are erudite and meticulous–they write with care and precision. Their book adopts the voice of a wise elder telling math stories to precocious children. It will interest readers of all ages who seek a window into current mathematics.”–Kenneth Ribet, University of California, Berkeley”Summing It Up ambitiously presents concepts of number theory, from the elementary to the advanced, for readers with only a knowledge of high school math and some calculus. With a crisp yet conversational style and excellent examples, Ash and Gross explain a great amount of interesting and important math.”–James Pommersheim, coauthor of Number Theory: A Lively Introduction with Proofs, Applications, and Stories”A well-motivated and fun introduction to some very deep topics in mathematics, Summing It Up takes readers on a mathematics tour starting with very classical material and building towards some amazing recent work, with many highlights along the way. It can be read by those with many different levels of mathematical background. I thoroughly enjoyed it.”–Nathan Kaplan, University of California, Irvine About the Author Avner Ash is professor of mathematics at Boston College. Robert Gross is associate professor of mathematics at Boston College. They are the coauthors of Elliptic Tales: Curves, Counting, and Number Theory and Fearless Symmetry: Exposing the Hidden Patterns of Numbers (both Princeton). Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I have the earlier two books by Ash and Gross. I particularly like “Elliptic Tales,” and so was anticipating this introduction to modular forms. But this book doesn’t quite work.The book is divided into three parts — 1) finite sums, 2) infinite sums, and 3) modular forms. The recurring theme that binds together the three parts is expressing numbers as sums of squares and of calculating the number of ways this can be done. In this connection, Bernoulli numbers (and functions) allow the authors to segue from the first part to the second. The discussion of Bernoulli numbers, the Riemann zeta-function, and generating functions is woefully brief — considering the conceptual leap required from the first part to the second and considering that the target readership is assumed to have no background in complex analysis. But this is a minor quibble.The third part attempts to introduce modular forms and in this connection discusses SL2(Z), fundamental domains, q-expansions, dimensions of vector spaces of modular forms, Hecke operators, and L-functions. The chapter on applications includes some desultory discussion of partitions that doesn’t lead anywhere and a worked example for the number of ways 6 can be expressed as a sum of squares — yet which doesn’t really employ the topics mentioned above. It seems to be a mix of weak heuristic explanation coupled with one solitary example. If the target reader is a scientific layman, I doubt he’ll be able to follow the discussion. If a math undergrad, the discussion won’t be ample enough, nor will it be precise and structured enough.To be fair, the math is difficult. My contention is that anyone trying to make this palatable to a lay readership will fail. The books I would propose for coverage of modular forms would be:1) Modular Functions and Dirichlet Series in Number Theory, by Apostol,2) Elliptic Curves, Modular Forms, and their L-Functions, by Lozano-Robledo,3) Introduction to the Mathematics of Fermat-Wiles, by Hellegouarch,4) Introduction to Elliptic Curves and Modular Forms, by Koblitz, and5) A First Course in Modular Forms, by Diamond and Shurman.

⭐Avner Ash and Robert Gross have written three roughly a series books ‘Elliptic Tales’, ‘Fearless Symmetry:Exposing the Hidden Patterns of Numbers’ and in 2016 ‘Summing It Up: From One Plus One to Modern Number Theory’.‘Summing it Up’ is somewhat different and less technical than the prior books but gets down to some of the basics of the other two books and is more accessible to a broader audience. It may seem more elementary in some ways but in fact gets behind some of the calculations in Number Theory. The writing and ‘tricks’ remind me of books by Paul Nahin and especially ‘Interesting Integrals’ for showing how to calculate things in different ways and how to get around problems when the calculations you try become a mess or seeming dead-end.

⭐I liked this book more than other similar books I’ve tried. Reading it is like having someone explain some math to you personally. No matter how much (or little) you know, there’s probably something new in here for you to think about, unless perhaps if you’re mentioned in this book’s references or are a number theory researcher who would be entirely comfortable talking a piece of chalk in hand after some random person asked “Say, I’d like to know what a modular form is.” In particular if you’ve seen the recent Ramanujan movie you might like this as a slow-starting but then ever-accelerating walkthrough of some of the number theory in it, maybe. By the intermediate value theorem, there’s provably a moment at which you have to be learning something.

⭐This is an excellent book. It is really worth reading it. It contains very good material on Number Theory starting from basic notions till modular forms and elliptic curves. Some mathematical background is necessary to follow the proofs, which are presented mostly in full details and in a very transparent way. Every mathematician interested in this subject should read it. It is a shortcut to understand the difficulties of some classic problems in number theory. Summarizing, this is a serious maths book written by professionals in the area in a very friendly way, and, most important, there is no bs here.

⭐Good but not what I expected.

⭐A very friendly and original introduction to number theory……

⭐Fantastic book for a panoramic view about number theory.

⭐I thought this book would be more of a popsci type book and less of a math textbook

⭐This is the third book by Ash and Gross the first two being “Fearless Symmetry” and “Elliptic Tales”. Since I have both of those I had to have this. And it’s brilliant. All three books are concerned with mathematics and number theory and while their first two books skirted around modular forms this one actually has modular forms as its destination. It is an easy read – you can skip some of the maths if what you want is a general background on number theory, or, you can role up your sleeves and grab a notepad and pen and follow along. Highly recommended.

⭐The authors have produced an amazingly clear introduction to modular forms using very little maths beyond calculus. If you are interested in number theory get hold of this book – you won’t regret it and it may inspire you to tackle more advanced books on the same topic.

⭐A lot of new material that I haven’t seen in previous recreational maths books. Plenty of proper mathematics to get your teeth into.

⭐well written and methods easy to follow.

⭐Excellent modern approach introducing everything associated with “sum”

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