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

Original · EN

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 L²[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+ck¹/³), achieved by the (k-1)st integral of x¹⁻ᵏ/²Jₖ₋₂(a₁√x), where a₁ is the first positive root of the (k-2)nd Bessel function Jₖ₋₂ and as such, is asymptotically ck¹/³ 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+Ck¹/³) with some other constant C. In the forthcoming paper of J. Pintz (also to appear in the Turán-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.