Lectures on the Riemann Zeta Function

H. Iwaniec

ISBN 10: 1470418517 / ISBN 13: 9781470418519

Published by American Mathematical Society, US, 2014

About this Item

The Riemann zeta function was introduced by L. Euler (1737) in connection with questions about the distribution of prime numbers. Later, B. Riemann (1859) derived deeper results about the prime numbers by considering the zeta function in the complex variable. The famous Riemann Hypothesis, asserting that all of the non-trivial zeros of zeta are on a critical line in the complex plane, is one of the most important unsolved problems in modern mathematics.The present book consists of two parts. The first part covers classical material about the zeros of the Riemann zeta function with applications to the distribution of prime numbers, including those made by Riemann himself, F. Carlson, and Hardy-Littlewood. The second part gives a complete presentation of Levinson's method for zeros on the critical line, which allows one to prove, in particular, that more than one-third of non-trivial zeros of zeta are on the critical line. This approach and some results concerning integrals of Dirichlet polynomials are new. There are also technical lemmas which can be useful in a broader context.

Seller Inventory # LU-9781470418519

Bibliographic Details

• Title: Lectures on the Riemann Zeta Function
• Publisher: American Mathematical Society, US
• Publication Date: 2014
• Language: English
• ISBN 10: 1470418517
• ISBN 13: 9781470418519
• Binding: Paperback
• Condition: New
• Dimensions: 17.15W x 0.64H x 24.77D centimeters 17.15 centimeters width by 0.64 centimeters height by 24.77 centimeters depth
• Weight: 235 grams

About this title

Synopsis

The Riemann zeta function was introduced by L. Euler (1737) in connection with questions about the distribution of prime numbers. Later, B. Riemann (1859) derived deeper results about the prime numbers by considering the zeta function in the complex variable. The famous Riemann Hypothesis, asserting that all of the non-trivial zeros of zeta are on a critical line in the complex plane, is one of the most important unsolved problems in modern mathematics. The present book consists of two parts. The first part covers classical material about the zeros of the Riemann zeta function with applications to the distribution of prime numbers, including those made by Riemann himself, F. Carlson, and Hardy-Littlewood. The second part gives a complete presentation of Levinson's method for zeros on the critical line, which allows one to prove, in particular, that more than one-third of non-trivial zeros of zeta are on the critical line. This approach and some results concerning integrals of Dirichlet polynomials are new. There are also technical lemmas which can be useful in a broader context.

About the Author

H. Iwaniec , Rutgers University, Piscataway, NJ, USA.