The CFZ–CS ratios conjecture for averages of zeta-function ratios

Let $A$ and $B$ be sets of complex numbers with real parts smaller than $1/4$, and let $C$ and $D$ be sets of complex numbers with positive real parts smaller than $1/4$. Set $s=1/2+it$ and define

$$\mathcal R_{A,B,C,D}(T):=\int_0^\infty \psi\left(\frac t T\right) \frac{\prod_{\alpha\in A}\zeta(s+\alpha)\prod_{\beta\in B}\zeta(1-s+\beta)}{\prod_{\gamma\in C}\zeta(s+\gamma)\prod_{\delta\in D}\zeta(1-s+\delta)}\,dt.$$

Let $\mathcal B_{A,B,C,D}(s):=\sum_{n=1}^\infty I_{A,C}(n)I_{B,D}(n)n^{-s}$, where $I_{A,C}$ is defined by

$$\sum_{n=1}^\infty \frac{I_{A,C}(n)}{n^s}=\frac{\prod_{\alpha\in A}\zeta(s+\alpha)}{\prod_{\gamma\in C}\zeta(s+\gamma)}.$$

The CFZ–CS ratios conjecture. Suppose the sets $A,B,C,D$ are as above and the imaginary parts of all parameters are $O(T^{1-\xi})$ for some $\xi>0$. Then, for some $\eta>0$,

$$\mathcal R_{A,B,C,D}(T)=\int_0^\infty \psi\left(\frac t T\right)\sum_{U\subset A,\,V\subset B\atop |U|=|V|}\left(\frac{t}{2\pi}\right)^{-\sum_{\hat\alpha\in U}\hat\alpha-\sum_{\hat\beta\in V}\hat\beta}\mathcal B_{A-U+V^-,B-V+U^-,C,D}(1)\,dt+O(T^{1-\eta}).$$

This is the ratios conjecture originally formulated by Conrey, Farmer and Zirnbauer and studied by Conrey and Snaith; the paper relates it to arithmetic correlations and proves that both conjectures imply the same result. Its general status is not established.