Euler Product Models of L-Functions and the Distribution of Zeros and Primes

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The proposer recently constructed one parameter families of functions whose members approximate the Riemann zeta-function or other L-functions, and whose structure incorporates the fundamental features of these functions, such as their Euler products and functional equations. Each function in a family satisfies a Riemann hypothesis with finitely many possible exceptions. Moreover, when the parameter is not too large, the functions have approximately the same number of zeros as the zeta or L-function, the zeros are all simple, and consecutive zeros repel. One may therefore regard them as models of the zeta-function or L-function. In fact, if the Riemann hypothesis holds for an L-function, the zeros of functions in the corresponding family tend to those of the L-function as the parameter increases, and between zeros on the critical line the functions tend to twice the L-function. The main goal of this project is to investigate the new functions further in order to gain insight into the behavior of the Riemannn zeta-function and L-functions and into connections between their zeros and the prime numbers. One project is to investigate how well the models approximate L-functions in a transitional region close to the critical line. Another is to study the approximations for a particularly important range of the parameter. In several projects the proposer will use the models to try to understand problems that have so far resisted other treatments, such as how the zeros of the derivative of the zeta-function are distributed. The present line of inquiry gives finite Euler products a more prominent role than previously in analytic number theory, so another goal is to study such products anew, particularly their moments. This is quite difficult from the traditional point of view, but the prposer's recent work leads to a new approach. A second and separate set of projects aims at furthering our understanding of the distribution of primes and zeros of L-functions. One is to explore connections between the Gaussian Unitary Ensemble hypothesis, the distribution of primes, and mean values of the zeta-function. Another is to calculate the discrete number variance and other statistics for the zeros of the zeta-function. A third project is to develop discrete mean values formulas for long Dirichlet polynomials and use these to estimate the size of gaps between zeros of the zeta-function.The projects proposed all address fundamental problems in analytic number theory: the distribution of prime numbers, the distribution of zeros of L-functions, and the general behavior of L-functions. Prime numbers are the ultimate building blocks of arithmetic, and therefore much of mathematics, so understanding their properties is of basic importance. Many of these properties are encoded in the Riemann zeta-function and other L-functions, and that makes these important objects of study as well. Most of the projects proposed center on the investigation of functions recently constructed by the proposer to model L-functions and capture their basic features. The structure of these simpler functions makes it clear why they behave as they do, thereby providing insight into the behavior of the actual L-functions. The remaining projects study the primes, L-functions, and their connections more directly.