Multiplicative Number Theory

研究者将继续三个项目在乘法数论。 第一个涉及到推广R. L. Graham关于a/gcd（a，B）在a，B上的最大值位于一个有限的整数集合中. 反过来，这个问题有一个应用程序的组合问题的配置相交的算术级数，研究人员最近取得了很大的进展。 第二个项目涉及的问题，在“比较素数理论”，理论的微妙的不平等分布的素数在算术级数。 第三个研究是研究者关于“移位素数”分布的工作的延伸（集合{p+a：a）及有关算术函数如欧拉函数、约数和函数等的问题。本项目涉及数论领域的若干问题，研究涉及整数的数学问题，（方程、不等式、分布等）。 近年来，数论在信息论中的许多应用已经被发现，例如“不可破解”密码系统（RSA），数据压缩和数字数据错误检测（在互联网和卫星通信中很重要）。 素数如何分布的问题是数论中的中心问题之一，在上述应用中起着重要的作用。 研究者和他的同事们将研究几个有关素数分布的问题。 一组问题涉及不同算术级数中素数分布的微妙不平等，而另一组问题涉及重要算术函数（定义在正整数上的函数f（n）取决于n的素数分解）。 研究人员已经在这些问题上取得了实质性的进展，进一步的进展将大大有助于素数理论。

The investigator will continue three projects in multiplicative number theory. The first involves extending a famous conjecture of R. L. Graham on the maximum of a/gcd(a,b) over a,b lying in a finite set of integers. In turn this problem has an application to a combinatorial problem on configurations of intersecting arithmetic progressions, on which the investigator has made substantial recent progress. The second project deals with questions in "comparative prime number theory", the theory of subtle inequities in the distribution of prime numbers in arithmetic progressions. The third investigation is an extension of the investigator's work on the distribution of "shifted primes" (sets {p+a: a prime} for fixed nonzero a) and related questions about arithmetic functions such as Euler's totient function and the sum of divisors function.This project concerns several problems in the the area of number theory, the study of mathematical problems involving whole numbers (equations, inequalities, distributions, etc). In recent years numerous applications of number theory to information theory have been discovered, such as "unbreakable" cryptosystems (RSA), data compression and digital data error detection (important in Internet and satellite communications). Questions about how the prime numbers are distributed form one of the central topics in number theory, and play important roles in the aforementioned applications. The investigator and his colleagues will study several questions concerning the distribution of primes. One set of problems concerns subtle inequities in the distribution of primes in different arithmetic progressions, while another set involves questions about important arithmetic functions (functions f(n) defined on the positive integers which depend on the prime factorization of n). The investigator has already made substantial progress on such questions, and further advances will contribute significantly to the theory of prime numbers.