Weighted Erd(?)s-Kac Type Theorems On Number Fields

Erd(?)s-Kac theorem is a very important research content in probabilistic number theory.This paper mainly studies weighted Erd(?)s-Kac type theorems in number field.Let K be a quadratic field over the rational field¤,O_K be the ring of algebraic integers in K,and a_K(n)be the number of non-zero integral ideals in_KO with norm n,l~+.In this paper,we use the Erd(?)s-Kac method and the Selberg-Delange method improved by Wu[Math.Nachr.,2020,293:178-202]to discuss the Erd(?)s-Kac type theorems weighted bya_K(n)~landa_K(n ~2)~lin short intervals,respectively.We also get asymptotic formulae for the average behavior of a_K(n)~landa_K(n ~2)~lin short intervals.The first chapter of this paper mainly introduces the research background and main results.In the second chapter,we mainly introduce two problems,the prime ideal theory of algebraic integer rings and the analytic properties of the Dedekind zeta function.In the third chapter,the Selberg-Delange method,an analytical method to estimate the mean value of arithmetic functions,is presented.In this method,the relationship between Dirichlet series corresponding to arithmetic functions and zeta function is established at first,and the related problems are studied by using the analytic properties of zeta function.In the fourth chapter,the Erd(?)s-Kac method,a probabilistic method to study the mean value of arithmetic function,is introduced.This method regards arithmetic function as a random variable in a probability space,and studies the limit distribution of random variable in the probability space.In the last chapter,we combine the Selberg-Delange method and the Erd(?)s-Kac method to prove the Erd(?)s-Kac type theorems weighted bya _K(n)~landa_K(n ~2)~lof quadratic field in short intervals,respectively.We also give the mean estimate of a_K(n)~landa_K(n ~2)~lin short intervals by the analytic properties of zeta function.