Multiplicative Number Theory

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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.