Robert Vaughan

Affiliation: Pennsylvania State University

Email: rvaughan@math.psu.edu

Title Of Talk: Some problems of 'Partitio Numerorum': Hybrid expressions

Abstract: As is well known, stimulated by the celebrated paper of Hardy and Ramanujan [1918] on the partition paper, Hardy and Littlewood produced a series of equally seminal papers with the generic title "Some problems of 'Partitio Numerorum'". In the third paper [1922] of this series with the subtitle "On the expression of a number as a sum of primes" in addition to primes they also consider their new method as a heuristic leading to a range of conjectures on a spread of additive problems. Many of these concern hybrid questions in which the summands can be combinations of primes and powers. Perhaps the most famous of these was question of the number of representations of a large number as a sum of a prime and two squares. Following fundamental work of Hooley [1957] this was eventually solved by Linnik [1961 by the dispersion method, and a second proof was given by Elliott and Halberstam [1966] using Bombieri's theorem [1965]. However on many of the questions there has been very little progress. On the other hand a host of similar but different questions have been examined over the years through developments of the Hardy-Littlewood method with varying amounts of success. We will give a brief survey of some of these questions and describe two in more detail.

Last update made Sat Oct 6 15:27:10 EDT 2012.
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