Softcover ISBN:  9781470462574 
Product Code:  STML/92 
List Price:  $59.00 
MAA Member Price:  $47.20 
AMS Member Price:  $47.20 
eBook ISBN:  9781470462796 
Product Code:  STML/92.E 
List Price:  $59.00 
MAA Member Price:  $47.20 
AMS Member Price:  $47.20 
Softcover ISBN:  9781470462574 
eBook: ISBN:  9781470462796 
Product Code:  STML/92.B 
List Price:  $118.00 $88.50 
MAA Member Price:  $94.40 $70.80 
AMS Member Price:  $94.40 $70.80 
Softcover ISBN:  9781470462574 
Product Code:  STML/92 
List Price:  $59.00 
MAA Member Price:  $47.20 
AMS Member Price:  $47.20 
eBook ISBN:  9781470462796 
Product Code:  STML/92.E 
List Price:  $59.00 
MAA Member Price:  $47.20 
AMS Member Price:  $47.20 
Softcover ISBN:  9781470462574 
eBook ISBN:  9781470462796 
Product Code:  STML/92.B 
List Price:  $118.00 $88.50 
MAA Member Price:  $94.40 $70.80 
AMS Member Price:  $94.40 $70.80 

Book DetailsStudent Mathematical LibraryVolume: 92; 2020; 152 ppMSC: Primary 11
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.
ReadershipUndergraduate and graduate students interested in analytic number theory.

Table of Contents

Chapters

The Riemann zeta function

The Euler product

The functional equation

The explicit formulas in analytic number theory

The prime number theorem

Oscillation of $\pi (x)\mathrm {Li}(x)$

The prime number race

Exercises, hints, and selected solutions


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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.
Undergraduate and graduate students interested in analytic number theory.

Chapters

The Riemann zeta function

The Euler product

The functional equation

The explicit formulas in analytic number theory

The prime number theorem

Oscillation of $\pi (x)\mathrm {Li}(x)$

The prime number race

Exercises, hints, and selected solutions