Admissibility of the constant 1 in the weighted large sieve

Determine whether the constant 1 is admissible in place of the current coefficient in the weighted large sieve inequality, thereby improving the coefficient appearing in the denominator of the associated sieve bound.

Background

The paper derives a weighted large sieve inequality whose denominator contains the term c q(q+Q), with c=0.898678… in the paper’s formulation, and compares this with the classical coefficient 3/2 in the Montgomery–Vaughan result. The discussion notes that E. Preissmann improved the coefficient 3/2 to approximately 1.315 and states that the constant 1 is conjectured to be admissible.

Establishing admissibility of the constant 1 would yield a stronger weighted large sieve inequality and, according to the paper, would improve the numerical constant in the resulting upper bound for the number of twin primes in an interval.