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

TL;DR

This paper optimizes the polynomial used in the GPY method for small prime gaps by solving an eigenvalue problem, deriving explicit solutions, and constructing near-optimal functions to enhance the method's effectiveness.

Contribution

It reformulates the polynomial optimization as an eigenvalue problem, explicitly solves the differential equation, and constructs a simpler near-optimal polynomial for the GPY method.

Findings

Maximal eigenvalue expressed as a Bessel function integral

Explicit solutions for eigenfunctions and eigenvalues

Construction of a near-optimal polynomial for the GPY method

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--Yildirim 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-1)st integral of x^{1-k/2}J_{k-2}(a_1\sqrt{x}), where a_1 is the first positive root of the (k-2)nd Bessel function J_{k-2} and as such, is asymptotically ck^{1/3} with a well-known constant c. 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 (also to appear in the Tur\'an-100 Memorial Volume by de Gruyter), 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.