In his paper "The pair correlation of zeros and the zeta function",
Montgomery defines a function
$$F(\alpha,T) = \left(\frac{T}{2 \pi} \log T\right)^{1} \sum_{0 < \gamma, \gamma' < T} T^{i \alpha (\gamma'\gamma)} w(\gamma'\gamma)$$
where $w(u)=\frac{4}{4+u^2}$, and the sum is over pairs of imaginary parts $\gamma, \gamma'$ of nontrivial zeros of the Riemann Zeta function.
Then he says that heuristic arguments suggest that "$F(\alpha)=1+o(1)$ for $\alpha \geq 1$, uniformly in bounded intervals". My question is: what does that mean?
Can someone please reformulate this conjecture with quantifiers in the right order? Thanks.
For any $\epsilon > 0$ and for any finite interval $I \subset [1, \infty)$, there is a $T_{0}$ such that for all $T > T_{0}$ and all $\alpha \in I$ we have $F(\alpha,T)  1 \leq \varepsilon$. The meaning of the conjecture is that the Fourier transform $F(\alpha, T)$ of the pair correlation of zeros up to height $T$, of the Riemann zetafunction converges to a limit $F(\alpha)$ as $T$ goes to infinity. You can invert the Fourier transform, and then read off the conjecture as saying that for any smooth Schwartz class function $f$, the limit $$ \lim_{T \rightarrow \infty} \frac{1}{N(T)}\sum_{0 < \gamma, \gamma' < T} f(\log T (\gamma  \gamma')) $$ exists, where $N(T)$ is the number of zeros up to $T$. Moreover the limit is an explicit linear functional of $f$. It coincides with the similar functional that one gets from considering the pair correlation of the eigenvalues of random GUE matrices. For better or for worse a lot of ink has been spilled on this last observation.
The importance of the conjecture is that it immediately implies that the normalized gaps between zeros of the Riemann zetafunction tend to get arbitrarily small. With a bit more work this then implies that there are no Siegel zeros. This was in fact Montgomery's original reason for considering the pair correlation of the zeros.
P.S: In your first display you should be dividing by $T \log T$ and not $T / \log T$, since $N(T) \sim (1/2\pi) T \log T$.

3$\begingroup$ In your reformulation (the displayed limit), you want to dilate $f$. $\endgroup$ – Lucia Nov 19 '17 at 19:29

$\begingroup$ Thanks. I corrected the normalization of the function $F$ in my post. Do you have a reference for the fact that the conjecture implies that there are no Siegel zeros? $\endgroup$ – Joël Nov 20 '17 at 19:57

3$\begingroup$ Joēl: Montgomery's paper gives the idea of why the conjecture implies no Siegel zeros  essentially all the small zeros of the corresponding Dedekind zeta function lie on the critical line, and in an almost arithmetic progression. Since these zeros include the zeros of $\zeta(s)$, knowing that two zeros of $\zeta$ can get very close gives a contradiction. The definitive treatment of this idea is in Conrey & Iwaniec (appeared in Acta Arith.) see arxiv.org/abs/math/0111012 . $\endgroup$ – Lucia Nov 20 '17 at 20:04