Problems in Analytic Number Theory

点击翻译按钮获取中文摘要

One of the most enduring problems of prime number theory is to determinehow many elements remain when a set of numbers is sieved. There is a largeliterature 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 provedto be optimal. By starting with a simple sieving situation and then movingincrementally to more complicated configurations, it is hoped that optimalbounds can be found, accompanied by proofs of optimality. The past workon the PI on the pair correlation of the zeros of the Riemann zeta functionhas been interpreted as providing evidence that the zeros are spectralin nature. The Pair Correlation Conjecture itself is equivalent to anassertion concerning the mean square distribution of primes in short intervals.In new work with k. Soundararajan, it is proposed to extend the second momentheuristics to other moments, and hence develop heuristics concerning thedistribution function of primes in short intervals. It is hope that thisnew 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 puzzleto mathematicians for many centuries. In the early 20th century, newideas were introduced, which allowed one to deal with sieving for primesas a problem of linear programming. This led to many new results, buteven today the linear programming extremals remain to be found in mostsituations. By starting with a simple situation and moving incrementallyto more complicated ones, it is hoped that it will at last bepossible to locate the extremal configurations. Heuristics concerningthe distribution of primes in short intervals can be developed from theHardy--Littlewood prime k-tuple conjecture, and the insights gained fromsuch reasoning has an impact on other aspects of prime number theory,including the famous Riemann Hypothesis which dates from 1860.