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.
H. Iwaniec, Rutgers University, Piscataway, NJ, USA.
Classical topics Panorama of arithmetic functions Sums of basic arithmetic functions 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 ?(x) The zero-free region and the PNT Approximate functional equations The Dirichlet polynomials Zeros off the critical line Zeros on the critical line 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 Smooth bump functions The gamma function Bibliography Index