Szemeredi's theorem - 30 views
http://in-theory.blogspot.com/2006_05_28_archive.html in theory Saturday, June 03, 2006 Szemeredi's theorem Szemeredi's theorem on arithmetic progressions is one of the great triumphs of the "Hung...
Are there infinitely many prime pairs with given even difference? Most mathematicians think so. Using a strong arithmetic hypothesis, Goldston, Pintz and Yildirim have recently shown that there are infinitely many pairs of primes differing by at most sixteen.There is extensive numerical support for the prime-pair conjecture (PPC) of Hardy and Littlewood [G.H. Hardy, J.E. Littlewood, Some problems of ‘partitio numerorum’. III: On the expression of a number as a sum of primes, Acta Math. 44 (1923) 1–70 (sec. 3)] on the asymptotic behavior of π2r(x), the number of prime pairs with p≤x. Assuming Riemann’s Hypothesis (RH), Montgomery and others have studied the pair-correlation of zeta’s complex zeros, indicating connections with the PPC. Using a Tauberian approach, the author shows that the PPC is equivalent to specific boundary behavior of a function involving zeta’s complex zeros. A certain hypothesis on equidistribution of prime pairs, or a speculative supplement to Montgomery’s work on pair-correlation, would imply that there is an abundance of prime pairs.
