| Siegel’s theorem is an important theorem in number theory,and its different equivalent forms involve important elements such as lower bound estimates and zeros of Dirichlet L-functions as well as class numbers of number fields.After the theorem was improved and extended by mathematicians such as Walfisz,Tatuzawa and Brauer,it has been widely used in algebraic number theory and analytic number theory,especially in the study of the class number of imaginary quadratic fields and the prime number theorem of arithmetic progression.The study of the history of Siegel’s theorem not only provides an understanding of the development of the theorem and the evolution of the ideas in it,but also helps to better grasp its applications in modern mathematics,so it has theoretical value and practical significance and it’s worth doing this.Based on reading a large amount of primary and research literature,this dissertation explores the intellectual origin,the development process and applications of Siegel’s theorem by taking the evolution of the content of Siegel’s theorem as a clue.And this study uses a combination of research methods such as documentary research method,chronological method,conceptual analysis method and sociological method.The main findings and conclusions obtained in this dissertation are as follows.1.This thesis explores the background and sources of thought behind the formulation of Siegel’s theorem,and clarifies the different forms of Siegel’s theorem.Landau,Heilbroon and other mathematicians proved Gauss ’s conjecture that the number of classes of imaginary quadratic fields tended to infinity,and their work was inspirational to Siegel.He continued to study the class number problem based on the previous work and proposed Siegel’s theorem in the process.The original form of Siegel ’s theorem was related to the lower bound of the Dirichlet L-function,which can be equivalently transformed into the lower bound of the class number,and later an equivalent form of the theorem for the zeros of the Dirichlet L-function was given.2.The different methods of proving Siegel’s theorem are illustrated by the examples of its elementary proofs.The classical proofs of Siegel’s theorem were given by Esterman and Goldfield respectively,with the former utilizing Taylor ’s theorem and Cauchy’s inequality and the latter utilizing a corollary of Dirichlet ’ s class number formula.Moreover,Pintz derived the equivalence between the two forms of Siegel’s theorem while proving it,which was the first derivation from the non-zero region of L(s,X)to the lower bound of L(1,X).3.This thesis discusses the contents of the Siegel-Tatuzawa theorem and the impetus of this improvement to the class number problem of quadratic fields.Tatuzawa gave the specific value of the invalid constant in Siegel ’s theorem in the presence of a possible exception,which facilitated the calculation of the discriminant of imaginary quadratic field with a fixed class number,but the final solution of Gauss ’s conjecture about the class number of such fields depended on a new approach concerning elliptic curves.Besides,the Siegel-Tatuzawa theorem can also give conclusions about elements such as fundamental units and regulators in the class number problem of real quadratic fields.4.This thesis introduces the Brauer-Siegel theorem and its applications to the class number problems of algebraic number fields.Brauer successively extended Siegel’s theorem in the case of quadratic fields to arbitrary fixed-order number fields and to sequences of regular fields satisfying certain conditions,providing a powerful reference for the study of the class numbers of different types of algebraic number fields.5.The important applications of Siegel ’ s theorem and Siegel-Walfisz theorem in problems for prime numbers are analyzed.Building on related work by Page,Walfisz used Siegel’s theorem to obtain a consistent remainder estimate for the prime number theorem of arithmetic progression with respect to tolerances,which was a result that occupied an important place in the proof of Goldbach’s conjecture about odd numbers. |
