xiv Foreword

provided that the right-hand integral exists. (Experts will recognize the

right-hand side as the formula for integration by parts for the Riemann–

Stieltjes integral, but defining the left-hand side in this manner allows us to

avoid assuming any knowledge of Riemann–Stieltjes integration.) We will

often apply partial summation in the following form, which is straightfor-

ward to verify directly: Suppose that a and b are real numbers with a ≤ b

and that we are given complex numbers an for all natural numbers n with

a n ≤ b. Put S(t) :=

∑

an≤t

an. If f is piecewise continuous on [a, b],

then

an≤b

anf(n) =

b

a

f(t) dS(t).

In order to paint an accurate portrait of the mathematical landscape

without straying off point, it has been necessary on occasion to state certain

theorems without proof; such results are marked with a star ( ). For some

of these results, proofs are sketched in the corresponding chapter exercises.

Acknowledgements

There are many people without whom this book could not have been written

and many others without whom this book would not be worth reading.

Key members of the first group include my middle and high-school teach-

ers Daniel Phelon, Sharon Bellak, and Jeff Miller. It is thanks to their tire-

less efforts that I was prepared to attend the Ross Summer Mathematics

Program at Ohio State University in 1998. There Arnold Ross, assisted by

my counselor Noah Snyder and my seminar instructor Daniel Shapiro, im-

pressed upon me the importance of grappling with mathematical ideas for

oneself. I regard this as the most important lesson I have learned so far on

my mathematical journey. As an undergraduate, I was the fortunate recip-

ient of generous mentoring from Andrew Granville and Matt Baker, and I

had the privilege of attending A. J. Hildebrand’s 2002 REU in number the-

ory. My subsequent graduate experience at Dartmouth College ranks as one

of the happiest times of my life, due in large measure to the wise guidance

of my advisor, Carl Pomerance.

My family — my father Lawrence, my mother Lolita, and my brother

Michael — has done so much for me over the years that it would be impos-

sible (and inappropriate!) for me to express the extent of my appreciation

in this brief space. Another friend for whom I am grateful beyond words is

Susan Roth, who for the last decade has accompanied me on many of my

(mis)adventures in genre television.

Mits Kobayashi cheerfully donated his time to prepare many of the fig-

ures included in the text. Both he and Enrique Trevi˜ no pointed out several