## Project Details

### Description

The overall plan of this project is to find significant connections between the additive and the multiplicative structures of the integers. In recent years more and the additive structure has found sophisticated uses in multiplicative problems, as has the the multiplicative structure been used to shed light on the additive structure. In particular Professor Vaughan will consider a number of open problems about these related structures such as the distribution of general sequences in arithmetic progressions, Waring's problem, generalizations of the Dirichlet divisor problem and Redheffer's matrix.

This is a project in Number theory. Since the beginning of science, mathematicians have studied the properties of the simple counting numbers, 1, 2, 3, 4.... Since these numbers can always be added, they have a strong additive structure. There are many applications of additive number theory to communications, probability, and other branches of mathematics and science. Since the integers can be multiplied, they also have a multiplicative structure. This multiplicative structure has been used in geometry, physics and is at the heart of most computer security systems. After 3000 years of study, we know a great deal about both of these structures. Surprisingly much less is known about the interactions of these two structures. Historically mathematicians greatest problems and greatest triumphs have hinged on understanding the interaction of addition and subtraction.

Status | Finished |
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Effective start/end date | 7/15/99 → 6/30/02 |

### Funding

- National Science Foundation: $78,500.00