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 topicsPanorama of arithmetic functionsSums of basic arithmetic functionsTchebyshev's prime seedsElementary prime number theoremThe Riemann memoirThe analytic continuationThe functional equationThe product formula over the zerosThe asymptotic formula for N(T)The asymptotic formula for ?(x)The zero-free region and the PNTApproximate functional equationsThe Dirichlet polynomialsZeros off the critical lineZeros on the critical lineThe critical zeros after LevinsonIntroductionDetecting critical zerosConrey's constructionThe argument variationsAttaching a mollifierThe Littlewood lemmaThe principal inequalityPositive proportion of the critical zerosThe first moment of Dirichlet polynomialsThe second moment of Dirichlet polynomialsThe diagonal termsThe off-diagonal termsConclusionComputations and the optimal mollifierSmooth bump functionsThe gamma functionBibliographyIndex