Iwaniec H. Lectures on the Riemann Zeta Function

New York: American Mathematical Society, 2014. — viii, 120 p. — (University lecture series; vol. 62). — ISBN 978-1-4704-1851-9.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.Preface
Part 1. Classical Topics
Panorama of Arithmetic Functions
The Euler–Maclaurin Formula
Tchebyshev’s Prime Seeds
Elementary Prime Number Theorem
The Riemann Memoir
The Analytic Continuation
The Functional Equation
The Product Formula over the Zeros
The Asymptotic Formula for N(T)
The Asymptotic Formula for ψ(x)
The Zero-free Region and the PNT
Approximate Functional Equations
The Dirichlet Polynomials
Zeros off the Critical Line
Zeros on the Critical Line
Part 2. The Critical Zeros after Levinson
Introduction
Detecting Critical Zeros
Conrey’s Construction
The Argument Variations
Attaching a Mollifier
The Littlewood Lemma
The Principal Inequality
Positive Proportion of the Critical Zeros
The First Moment of Dirichlet Polynomials
The Second Moment of Dirichlet Polynomials
The Diagonal Terms
The Off-diagonal Terms
Conclusion
Computations and the Optimal Mollifier
Appendix A. Smooth Bump Functions
Appendix B. The Gamma Function
Bibliography
Index