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A large number of problems in number theory can be reduced to statements about L-functions. In this thesis, we study L-functions at the edge of the critical strip, and relate these to a variety of objects of arithmetic interest.
A large number of problems in number theory can be reduced to statements about L-functions. In this thesis, we study L-functions at the edge of the critical strip, and relate these to a variety of objects of arithmetic interest.
In analytic number theory, and increasingly in other surprising places, L-functions arise naturally when describing algebraic and geometric phenomena. For example, when attempting to prove the Prime Number Theorem the values of L-functions on the one-line played a crucial role. In this thesis we discuss the theory of L-functions in two different settings. In the classical context we provide results which give estimates for the size of a general L-function on the right edge of the critical strip, that is complex numbers with real part one. We also provide a bound for the number of zeros for the classical Riemann zeta function inside the critical strip commonly referred to as a zero density estimate. In the second setting we study L-functions over the polynomial ring A, which is all polynomials with coefficients in a finite field of size q. As A and the ring of integers have similar structure, A is a natural candidate for analyzing classical number theoretic questions. Additionally, the truth of the Riemann Hypothesis (RH) in A yields deeper unconditional results currently unattainable over the integers. We will focus on the distribution of values of specific L-functions in two different places: On the right edge of the critical strip, that is complex numbers with real part one, and inside of the critical strip, meaning the complex numbers will have real part between one half and one.
This volume develops methods for proving the non-vanishing of certain L-functions at points in the critical strip. It begins at a very basic level and continues to develop, providing readers with a theoretical foundation that allows them to understand the latest discoveries in the field.
This volume develops methods for proving the non-vanishing of certain L-functions at points in the critical strip. It begins at a very basic level and continues to develop, providing readers with a theoretical foundation that allows them to understand the latest discoveries in the field.
These notes present recent results in the value-distribution theory of L-functions with emphasis on the phenomenon of universality. Universality has a strong impact on the zero-distribution: Riemann’s hypothesis is true only if the Riemann zeta-function can approximate itself uniformly. The text proves universality for polynomial Euler products. The authors’ approach follows mainly Bagchi's probabilistic method. Discussion touches on related topics: almost periodicity, density estimates, Nevanlinna theory, and functional independence.
For thirty years, the biennial international conference AGC T (Arithmetic, Geometry, Cryptography, and Coding Theory) has brought researchers to Marseille to build connections between arithmetic geometry and its applications, originally highlighting coding theory but more recently including cryptography and other areas as well. This volume contains the proceedings of the 16th international conference, held from June 19–23, 2017. The papers are original research articles covering a large range of topics, including weight enumerators for codes, function field analogs of the Brauer–Siegel theorem, the computation of cohomological invariants of curves, the trace distributions of algebraic groups, and applications of the computation of zeta functions of curves. Despite the varied topics, the papers share a common thread: the beautiful interplay between abstract theory and explicit results.