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We work out the optimization problem, initiated by K. Soundararajan, for thechoice of the underlying polynomial P used in the construction of the weightfunction in the Goldston--Pintz--Yildirim method for finding small gaps betweenprimes. First we reformulate to a maximization problem on L^2[0,1] for aself-adjoint operator T, the norm of which is then the maximal eigenvalue of T.To find eigenfunctions and eigenvalues, we derive a differential equation whichcan 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 isthe first positive root of the (k-2)nd Bessel function J_{k-2} and as such, isasymptotically ck^{1/3} with a well-known constant c. As this naturally givesrise to a number of technical problems in the application of the GPY method, wealso construct a polynomial P which is a simpler function yet it furnishes anapproximately optimal extremal quantity, 4/(k+Ck^{1/3}) with some otherconstant C. In the forthcoming paper of J. Pintz (also to appear in theTur\'an-100 Memorial Volume by de Gruyter), it is indeed shown how thisquasi-optimal choice of the polynomial in the weight finally can exploit theGPY method to its theoretical limits.

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