Statement of the pair correlation conjecture

Question

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 non-trivial 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.

Answer 1

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 zeta-function 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 zeta-function 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 $N(T) \sim (1/2\pi) T \log T$.