Study of Riemann’s Zeta Function

This book is for readers of our first book who want to further advance their knowledge of the zeta function and the theory of prime numbers. The focus here is on work done in the 1900-1932 period.

For both books, our target audience is the math “hobbyist” – someone who has a good understanding of complex analysis but is not a math “guru” who eats, sleeps and breathes number theory and complex analysis. Our goal is to provide proofs that contain enough detail for the hobbyist reader to follow the proofs from beginning to end. If you need a refresher to get back to the hobbyist level, you might consider our second book, Complex Analysis: a Self-Study Guide.

There are almost an unlimited number of topics that could be included in this book. We chose a balance of: (1) “things you should know” (important theory that is related to, but independent from, the zeta function) plus (2) detailed proofs of several interesting advances in zeta-related theory. We then end with a chapter showing techniques for locating zeta zeros.

We hope you can use Amazon’s “read sample” feature to see our Table of Contents and Preface. If not, we briefly describe the contents below (full versions available at our web site).

Chapters 1 through 5 (Things You Should Know).

• The Riemann–Stieltjes integral (useful in computing integrals and sums involving step functions).
• The big O, big-Omega and related relationships.
• Dirichlet series – convergence and abscissa of convergence.
• Chebyshev and related functions, including Chebyshev’s important efforts (even though unsuccessful) in proving the PNT. Chebyshev’s ideas remain important and are well worth your study.
• The Pigeonhole Principle / Dirichlet’s Theorem (an important tool for large-scale number theory problems).

Chapters 6 through 12 (Zeta Function Properties).

• A good quality zeta approximation, plus estimates (conditional on the RH) for Chebyshev and related functions.
• Properties of N(T) beyond those found in our first book, including the error term S(T) and gaps between zeta zeros.
• Growth rate of zeta along all vertical lines as Im(s) goes to infinity.
• Properties of the Chi function and Hardy’s Z Function, essential functions for locating zeros on the critical line.
• Using Hardy’s Z Function, we show zeta has infinitely many zeros on the critical line.
• Mean value theorems. We apply our results in the next chapter.
• For any positive epsilon (no matter how small), we show that all but an “infinitesimal proportion” of zeta’s non-trivial zeros lie no further than epsilon away from the critical line.

Chapters 13 through 15 (Oscillation of Error Terms – Littlewood’s Theorem).

• We introduce the topic and discuss why all the great mathematicians thought li(x) > pi(x) for all x.
• We show the important work by Schmidt in obtaining several big-Omega results.
• After Schmidt, Littlewood was left with the very difficult task of proving the existence of infinitely many crossings of pi(x) and li(x), assuming the RH is true. The difficulty is the loss of “wiggle room” in the error term for the difference li(x) – pi(x). Littlewood’s proof is a tour de force.

Chapters 16: Locating Zeta Zeros.
Using Hardy’s Z Function, we show two different techniques for locating zeta zeros on the critical line: (1) Euler-Maclaurin summation (briefly discussed), and (2) the Riemann-Siegel Formula (our focus).

Our hope is that this book will give the reader a much-improved feel for the properties of the zeta function. We also believe there is much to be gained by carefully studying the proofs and techniques of the great mathematicians.