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

Amazingly, this slim book will take you from the basics to the frontiers on Riemann’s zeta function." - Tamás Waldhauser, ACTA Sci. Math"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

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

Amazingly, this slim book will take you from the basics to the frontiers on Riemann’s zeta function." - Tamás Waldhauser, ACTA Sci. Math"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

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

Amazingly, this slim book will take you from the basics to the frontiers on Riemann’s zeta function." - Tamás Waldhauser, ACTA Sci. Math"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