Have you ever wondered about the explicit
formulas in analytic number theory? This short book provides a
streamlined and rigorous approach to the explicit formulas of Riemann
and von Mangoldt. The race between the prime counting function and the
logarithmic integral forms a motivating thread through the narrative,
which emphasizes the interplay between the oscillatory terms in the
Riemann formula and the Skewes number, the least number for which the
prime number theorem undercounts the number of primes. Throughout the
book, there are scholarly references to the pioneering work of Euler.
The book includes a proof of the prime number theorem and outlines a
proof of Littlewood's oscillation theorem before finishing with the
current best numerical upper bounds on the Skewes number.
This book is a unique text that provides all the mathematical
background for understanding the Skewes number. Many exercises are
included, with hints for solutions. This book is suitable for anyone
with a first course in complex analysis. Its engaging style and
invigorating point of view will make refreshing reading for advanced
undergraduates through research mathematicians.
Readership
Undergraduate and graduate students interested in
analytic number theory.