Summing it up: from one plus one to modern number theory (Q2810188)

Readers of the books [the authors. Fearless symmetry. Exposing the hidden patterns of numbers. Princeton, NJ: Princeton University Press (2006; Zbl 1111.11001)] and [the authors. Elliptic tales. Curves, counting, and number theory. Princeton, NJ: Princeton University Press (2012; Zbl 1256.11036)] will not need this review to be talked into buying ``Summing it up'' from 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. The first part of the present book deals with sums of squares, proceeds to Waring's problem, and ends with formulas for sums of powers involving Bernoulli numbers and the technique of Euler-Maclaurin summation. This part is accessible for all readers familiar with basic calculus. The second part deals with infinite sums, that is, series. It presents the determination of the values of the zeta function at even positive integers and introduces generating functions. The third and main part deals with modular forms; keywords are transformation properties, growth conditions, the dimensions of spaces of modular forms and cusp forms, Eisenstein series, congruence subgroups, Hecke operators and Galois representations.NEWLINENEWLINEAs in the authors' other books, the discussion is nontechnical and informal, so readers should not expect a thorough introduction to modern number theory. In order to get the most out of this book, one should be familiar with the basics of analytic number theory and elliptic curves, although some sections may be read with less background.