Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the optimal weight function in the Goldston-Pintz-Yildirim method for
  finding small gaps between consecutive primes   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.

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The categories of this paper are: supr-con
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The categories of this paper are: q-alg
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas