Dirichlet Series and Analysis on Polydiscs

The project has developed a research area that spans analytic number theory, operator theory, and harmonic and complex analysis. The central goal of this research is to achieve an improved understanding of how the prime numbers are distributed. During 2014, workers in the project solved an almost 30 year old problem concerning so-called GCD matrices, closely connected with the growth of the Riemann zeta function on the critical line. In 2015, this work led the project to a new lower bound for the growth of the Riemann zeta function, which is the first essential improvement since 1977. In 2017, a similar technique was used to obtain improved estimates for argument of the zeta function, which is related to the distribution of the nontrivial zeros and hence, via a classical duality, to the distribution of the prime numbers. The project has continued working to enhance our understanding of the Riemann zeta function, and further improvements may be expected in future the projects. Other results deal with our understanding of basic operators acting on spaces of Dirichlet series. An example of such an operator is a multiplicative version av Hilbert's classical matrix. It appears naturally as the matrix of a certain integral operator involving the Riemann zeta function. In 2016, the project finished a first study of another kind of basic operators, namely so-called Volterra operators, displaying a novel and unconventional interaction between spaces of holomorphic functions and number theory. More recently, the project has studied contractive inequalities with applications to improved estimates for pseudomoments of the Riemann zeta function.

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The proposal addresses major problems concerning Hardy spaces of Dirichlet series, boundary limits of Dirichlet series, the interplay between complex analytic methods, probabilistic techniques, and analytic number theory in the study of Dirichlet series a nd Dirichlet polynomials, function theory in polydiscs, and harmonic analysis and operator theory on the infinite-dimensional torus. The proposal will - search for novel concepts and methods required to solve some of the basic problems of operator-related function theory and harmonic analysis on the infinite-dimensional torus - direct efforts to significantly strengthen the interplay between the different research directions involved in the project.