Fourier optimization and Montgomery's pair correlation conjecture (Q6622395)

The Montgomery function \(F(\alpha, T)\) for \(\alpha\in{\mathbb{R}}\) and \(T\geq15\), is defined by \N\[\NF(\alpha, T):=\frac{1}{N(T)}\sum_{0<\gamma,\gamma'\leq T}T^{i\alpha(\gamma-\gamma')}w(\gamma-\gamma'),\N\]\Nwhere \(N(T)\) denotes the number of non-trivial zeros \(\rho\) of the Riemann zeta function \(\zeta(s)\) with ordinates \(\gamma\) in the interval \([0,T]\) and \(w(u):=\frac{4}{4+u^{2}}\).\N\NIn this paper under review, the authors give in their main result (see Theorem 1) an upper and lower bounds for (the average value of \(F(\alpha,T)\)) \N\[\N\frac{1}{l}\int_{b}^{b+l}F(\alpha,T)d\alpha\N\]\Nunder the Riemann hypothesis, for \(b\geq1\) and large \(l\). A refinement of their result is given in Theorem 3 where they assume the generalized Riemann hypothesis for Dirichlet \(L\)-functions. Furthermore, they conclude that such an average value, that is conjectured to be 1, lies between 0.9303 and 1.3208.