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University Lecture Series
2014; 119 pp; softcover
List Price: US$40
Member Price: US$32
Order Code: ULECT/62
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.
Undergraduate and graduate students and research mathematicians interested in number theory and complex analysis.
"The book under review presents a number of essential research directions in this area in a friendly fashion. It focuses mainly on two questions: the location of zeros in the critical strip (the strip on the complex plane consisting of numbers with real part between 0 and 1) and the proportion of zeros on the critical line (the line at the center of the strip, consisting of numbers with real part 1/2). ... The book is quite technical, and readers need a basic knowledge in complex function theory and also analytic number theory to follow the details. The chapters are short, well-motivated, and well written; there are several exercises. Thus, the book can serve as a source for researchers working on the Riemann zeta-function and also to be a good text for an advanced graduate course."
-- MAA Reviews
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