On the optimal weight function in the Goldston–Pintz–Yıldırım method for finding small gaps between consecutive primes

Abstract

We work out the optimization problem, initiated by K. Soundararajan, for the choice of the underlying polynomial P used in the construction of the weight function in the Goldston-Pintz-Yıldırım method for finding small gaps between primes.First we reformulate to a maximization problem on L 2 [0, 1] for a self-adjoint operator T , the norm of which is then the maximal eigenvalue of T .To find eigenfunctions and eigenvalues, we derive a differential equation which can be explicitly solved.The aimed maximal value is S(k) = 4/(k + ck 1/3 ), achieved by the k -, where α 1 ∼ ck 1/3 is the first positive root of the k -2 nd Bessel function J k-2 .As this naturally gives rise to a number of technical problems in the application of the GPY method, we also construct a polynomial P which is a simpler function yet it furnishes an approximately optimal extremal quantity, 4/(k + Ck 1/3 ) with some other constant C. In the forthcoming paper of J. Pintz [8] it is indeed shown how this quasi-optimal choice of the polynomial in the weight finally can exploit the GPY method to its theoretical limits.

Locations

• Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) - View - PDF
• arXiv (Cornell University) - View - PDF
• CiteSeer X (The Pennsylvania State University) - View - PDF
• De Gruyter eBooks - View