Problems in Analytic Number Theory

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

One of the most enduring problems of prime number theory is to determine how many elements remain when a set of numbers is sieved. There is a large literature on this topic, originating in seminal papers of Viggo Brun in 1914, but even today the best known bounds are either not optimal or not proved to be optimal. By starting with a simple sieving situation and then moving incrementally to more complicated configurations, it is hoped that optimal bounds can be found, accompanied by proofs of optimality. The past work on the PI on the pair correlation of the zeros of the Riemann zeta function has been interpreted as providing evidence that the zeros are spectral in nature. The Pair Correlation Conjecture itself is equivalent to an assertion concerning the mean square distribution of primes in short intervals. In new work with k. Soundararajan, it is proposed to extend the second moment heuristics to other moments, and hence develop heuristics concerning the distribution function of primes in short intervals. It is hope that this new information, when interpreted in terms of zeros of the zeta function, will provide further insights concerning the distribution of the zeros, including the Riemann Hypothesis. The seemingly irregular distribution of prime numbers has been a puzzle to mathematicians for many centuries. In the early 20th century, new ideas were introduced, which allowed one to deal with sieving for primes as a problem of linear programming. This led to many new results, but even today the linear programming extremals remain to be found in most situations. By starting with a simple situation and moving incrementally to more complicated ones, it is hoped that it will at last be possible to locate the extremal configurations. Heuristics concerning the distribution of primes in short intervals can be developed from the Hardy--Littlewood prime k-tuple conjecture, and the insights gained from such reasoning has an impact on other aspects of prime number theory, including the famous Riemann Hypothesis which dates from 1860.