Review of

**Summing It Up: From One Plus One to Modern Number Theory**, by Avener Ash and Robert Gross, Princeton University Press, Princeton, New Jersey, 2016. 248 pp., $27.95 (hardbound). ISBN 978-0-691-17019-0.

Five out of five stars

This book is
one of the set of math books where the claims on the DJ are not justified by
the contents. The last paragraph on the front flap of the DJ is as follows:

“Appropriate for numbers novices as well as college
math majors,

**Summing It Up**delves into mathematics that will enlighten anyone fascinated by numbers.”
Although this makes it appear as if the book is one of
popular mathematics, there is nothing novice about it.

The first
section on finite sums contains material found in math major level courses in number
theory. Topics such as sums of squares, figurate numbers and quadratic residues
appear. The following section moves into much more advanced topics such as
lengthy sums and series, double and telescoping sums, Bernoulli numbers and
complex analytic functions. There is no sparing of the formula or proof of the
theorems here, when mathematics is needed to completely explain a concept, it
is used. The last section deals with modular forms and takes the reader into
topics such as elliptical curves Hecke operators and L-functions.

While this is a
very good book for people interested in advanced number theory, much of the
material is beyond the coverage of a math major course in number theory. It
will definitely not enlighten the person that has only a fascination with
numbers.

This book was made available for free for review purposes

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