2. The ratio formula: Theorem 4

In this section, we prove Theorem 4. Our starting point is an application of a general formula of Borodin-Olshanski-Strahov to the ACUE.

Theorem 6 (A Borodin-Olshanski-Strahov Formula for ACUE). For $N$ and $J$ positive integers, $v_1,\ldots,v_J$ complex numbers, and $u_1,\ldots,u_J$ complex numbers which are not $2N$ -th roots of unity with $v_i \neq u_j$ for all $i,j$ ,

(2.1) \begin{align} \mathbb{E}_{\textrm{ACUE}(N)} \Bigg [\frac{\prod _{j=1}^J \det (1+v_j g)}{\prod _{j=1}^J \det (1+u_j g)} \Bigg ] = \frac{1}{\det \left(\frac{1}{u_i-v_j}\right)} \det \left (\frac{1}{u_i-v_j} \mathbb{E}_{\textrm{ACUE}(N)} \left[\frac{\det (1+v_j g)}{\det (1+u_i g)}\right]\right ), \end{align}

where the determinants on the right-hand side are of $J \times J$ matrices, over the indices $1 \leq i,j \leq J$ .

Proof. This requires only minor modifications of formulas in [Reference Borodin, Olshanski and Strahov5]. Claims I and II of that paper show that if $\mu$ is a measure on $\mathbb{C}$ with finite moments and if a point process consisting of $N$ points $\{z_1,\ldots,z_N\}$ in $\mathbb{C}$ has a joint density given by

\begin{equation*} (\textrm {const.}) \prod _{1\leq i \lt j \leq N} |z_i-z_j|^2 \prod _{i=1}^N \mu (dz_i), \end{equation*}

then as a formal powers series

\begin{align*} \mathbb{E} \big [H(\alpha _1)\cdots H(\alpha _J) E(\beta _1)\cdots E(\beta _J) \big ] = \frac{1}{\det \left(\frac{1}{\alpha _i+\beta _j}\right)} \det \left (\frac{1}{\alpha _i+\beta _j} \mathbb{E} \big [H(\alpha _i) E(\beta _j)\big ]\right ), \end{align*}

where

\begin{equation*} H(\alpha ) \,:\!=\, \frac {1}{\prod _{j=1}^N \left(1-z_j \alpha ^{-1}\right)}, \quad E(\beta )\,:\!=\, \prod _{j=1}^N\left(1+z_j \beta ^{-1}\right). \end{equation*}

This is only claimed for a measure $\mu$ supported on $\mathbb{R}$ , but the proof applies with no change to measures supported on $\mathbb{C}$ , except that in the proof of Theorem 3.1 the moments $A_n = \int _{\mathbb{R}} x^n \mu (dx)$ must be replaced by $A_{n,m} = \int _{\mathbb{C}} z^n \overline{z}^m\, \mu (dz)$ and later in the proof $A_{\lambda _i + N-i + N-j}$ must be replaced by $A_{\lambda _i + N - i\,,\, N-j}$ .

The point process ACUE is induced by such a joint density where $\mu$ is a probability measure uniform on the $2N$ -th roots of unity in $\mathbb{C}$ . This identity may be seen to be true not just for formal powers series but for functions $H(\alpha )$ , $E(\beta )$ by considering the case $|\alpha _1|,\ldots,|\alpha _J| \gt 1$ (where all power series will converge absolutely) and then meromorphically continuing to all $\alpha _1,\ldots,\alpha _J$ .

Finally, we arrive at (2.1) simply by setting $\alpha _j = - u_j^{-1}$ , $\beta _j = v_j^{-1}$ and simplifying the resulting determinants.

The remainder of this section is therefore devoted to understanding the expectation which occurs on the right-hand side of (2.1), accomplished in Proposition 8 below.

Lemma 7. Consider a hook partition $(a,1^b)$ with $a\geq 1$ and $b \geq 0$ of length $b+1 \leq N$ . For the Schur polynomial $s_{(a,1^b)}$ associated with this partition in the variables $e(\vartheta _1),\ldots,e(\vartheta _N)$ of the $\textrm{ACUE}(N)$ , we have

\begin{equation*} \mathbb {E}_{\textrm {ACUE}(N)}\, s_{(a,1^b)} = \begin {cases} ({-}1)^b & \textrm {if}\quad a+b \equiv 0 \ (\textrm {mod}\ 2N) \\[5pt] 0 & \textrm {otherwise}. \end {cases} \end{equation*}

Proof. Label $\omega _j = e(\vartheta _j)$ so that for a partition $\lambda$ of length $\ell (\lambda ) \leq N$ ,

\begin{equation*} s_\lambda = \frac {\det \left(\omega _i^{\lambda _j + N -j}\right)}{\det \left(\omega _i^{N-j}\right)}, \end{equation*}

where if $\ell (\lambda ) \lt N$ we adopt the convention $\lambda _{\ell (\lambda )+1} = \cdots = \lambda _N = 0$ , and the determinants above are $N \times N$ .

Note that $\det \left(\omega _i^{N-j}\right) = \Delta (\omega _1,\ldots,\omega _N)$ . Hence from the definition (1.1) of the ACUE,

(2.2) \begin{align} \mathbb{E}_{\textrm{ACUE}(N)}\, s_\lambda = \frac{1}{N! \, (2N)^N} \sum _{t_1,\ldots,t_N} \det \big (e\big ((\lambda _j + N - j) t_i\big )\big ) \overline{\det \big (e\big ((N - j) t_i\big )\big )}, \end{align}

where each index $t_i$ is summed over the set $\left\{0, \tfrac{1}{2N},\ldots,\tfrac{2N-1}{2N}\right\}$ . Expanding each determinant into a sum over permutations mapping $\{1,\ldots,N\}$ to $\{1,\ldots,N\}$ one sees

\begin{align*} \det \big (e\big (\lambda _j + N - j) t_i\big )\big ) \overline{\det \big (e\big (N - j) t_i\big )\big )} = \sum _{\sigma, \pi \in S_N} ({-}1)^\sigma ({-}1)^\pi \prod _{i=1}^N e\big ((\lambda _{\sigma (i)} + N - \sigma (i)) t_i - (N - \pi (i)) t_i\big ). \end{align*}

Thus, (2.2) is

(2.3) \begin{align} \notag & = \frac{1}{N!} \sum _{\sigma,\pi \in S_N} ({-}1)^\sigma ({-}1)^\pi \textbf{1}\big [(\lambda _{\sigma (i)} - \sigma (i)) + \pi (i) \equiv 0 \ (\textrm{mod}\ 2N)\quad \textrm{for all}\; i\big ] \\ &= \sum _{\pi \in S_N} ({-}1)^\pi \textbf{1}\big [\lambda _{i} - i + \pi (i) \equiv 0 \ (\textrm{mod}\ 2N)\quad \textrm{for all}\; i\big ], \end{align}

where $\textbf{1}[\,\cdot \,]$ denotes an indicator function, taking the value $1$ or $0$ depending on whether the proposition inside is true or false.

In the special case that $\lambda = (a,1^b)$ , this sum has a simple evaluation. In that case, any nonvanishing summand will have $\pi$ satisfying

\begin{align*} a-1 + \pi (1) &\equiv 0 \ (\textrm{mod}\ 2N) \\ &\textrm{and}\\ 1 - 2 + \pi (2) &\equiv 0 \ (\textrm{mod}\ 2N) \\ &\vdots \\ 1 - (b+1) + \pi (b+1) &\equiv 0 \ (\textrm{mod}\ 2N)\\ &\textrm{and}\\ 0 - (b+2) + \pi (b+2) &\equiv 0 \ (\textrm{mod}\ 2N) \\ &\vdots \\ 0 - N + \pi (N) &\equiv 0 \ (\textrm{mod}\ 2N). \end{align*}

Since $1 \leq \pi (i) \leq N$ , the last $N-1$ of these equations force

\begin{equation*} \pi (b+2) = b+2, \; \ldots ., \; \pi (N) = N, \end{equation*}

\begin{equation*} \pi (2) = 1, \; \ldots ., \; \pi (b+1) = b. \end{equation*}

This forces

\begin{equation*} \pi (1) = b+1, \end{equation*}

and so at most one permutation $\pi$ makes a nonzero contribution to (2.3), and that contribution is nonzero if and only if $a+b \equiv 0 \ (\textrm{mod}\ 2N)$ , since $a + b = a - 1 + \pi (1)$ . Since in cycle notation this permutation is $\pi = (b+1,\, b, \, \ldots, \, 2, \, 1)$ , we have $({-}1)^\pi = ({-}1)^b$ , and this verifies the lemma.

Proposition 8. For $v$ any complex number and $u$ any complex number which is not a $2N$ -th root of unity,

\begin{equation*} \mathbb {E}_{\textrm {ACUE}(N)} \frac {\det (1+vg)}{\det (1+ug)} = \frac {1- u^N v^N}{1-u^{2N}}. \end{equation*}

Proof. We first consider $|u|\lt 1$ . From a series expansion, we have

(2.4) \begin{equation} \frac{\det (1+vg)}{\det (1+ug)} = \sum _{j=0}^N \sum _{k=0}^\infty ({-}1)^k e_j h_k v^j u^k, \end{equation}

where $e_j$ and $h_k$ are respectively elementary symmetric polynomials of degree $j$ and homogeneous symmetric polynomials of degree $k$ in the variables $e(\vartheta _1),\ldots,e(\vartheta _N)$ associated with $\textrm{ACUE}(N)$ . Note that

\begin{equation*} e_0 = h_0 = 1, \end{equation*}

while other terms can be expression in terms of Schur polynomials in the variables $e(\vartheta _1),\ldots,e(\vartheta _N)$ :

\begin{align*} e_j h_0 &= s_{(1^j)} \quad \textrm{for}\; 1 \leq j \leq N,\\ e_0 h_k &= s_{(k)} \quad \textrm{for}\; k \geq 1,\\ e_j h_k &= s_{(k+1,1^{j-1})} + s_{(k,1^j)} \quad \textrm{for}\; 1 \leq j \leq N-1, \, k\geq 1,\\ e_N h_k &= s_{(k+1,1^{N-1})} \quad \textrm{for}\; k\geq 1, \end{align*}

with the first two identities following from the combinatorial definition of Schur functions [Reference Stanley30, Sec. 7.10], and the last two from the Pieri rule [Reference Stanley30, Thm. 7.15.7]. From Lemma 7, it thus follows

\begin{equation*} \mathbb {E}_{\textrm {ACUE}(N)} e_j h_k = \begin {cases} 1 & \textrm {if}\; j = 0, \; k \equiv 0 \ (\textrm {mod}\ 2N), \\ ({-}1)^{N-1} & \textrm {if}\; j = N, \; k \equiv N \ (\textrm {mod}\ 2N), \\ 0 & \textrm {otherwise}. \end {cases} \end{equation*}

Hence from (2.4),

\begin{align*} \mathbb{E}_{\textrm{ACUE}(N)} \frac{\det (1+vg)}{\det (1+ug)} & = (1+ u^{2N} + u^{4N} + \cdots ) - (v^N u^N + v^N u^{3N} + v^N u^{5N} + \cdots ) \\ &= \frac{1-u^N v^N}{1-u^{2N}}, \end{align*}

for $|u| \lt 1$ and all $v$ . The result then follows by analytic continuation.

Thus we have:

Proof of Theorem 4. Apply Proposition 8 to Theorem 6.

3. The moment formula: Theorem 2

Our technique in proving Theorem 2 will be to condense the determinants in (1.5) by letting all $u_i\rightarrow 0$ . We begin with several lemmas that are useful for that purpose.

The following is a slight generalization of Lemma 1 of [Reference Medjedovic23].

Lemma 9 (Determinantal Condensation Identity). Take $q \leq J$ . For $f_1, f_2,\ldots,f_J$ functions (mapping $\mathbb{R}$ to $\mathbb{C}$ ) that are at least $q$ times continuously differentiable at the point $a$ ,

(3.1) \begin{equation} \lim _{u_1,\ldots,u_q \rightarrow a} \frac{1}{\Delta (u_q,\ldots,u_1)}\det \big (f_j(u_i)\big )_{i,j=1}^J = \det \begin{pmatrix} \Big (\frac{1}{(i-1)!} f_j^{(i-1)}(a)\Big )_{i \leq q,\;\, j \leq J} \\[10pt] \big (f_j(u_i)\big )_{q+1 \leq i \leq J,\;\, j \leq J}\end{pmatrix}, \end{equation}

where on the left-hand side the limit is taken in the order that first $u_1 \rightarrow u_2$ , then $u_2 \rightarrow u_3$ , …, $u_{q-1}\rightarrow u_q$ , and finally $u_q \rightarrow a$ .

Proof. We prove this identity by induction, viewing $\Delta (u_1) = 1$ for the $q = 1$ case, which then becomes trivial. Suppose then that (3.1) has been proved for a limit in $q-1$ variables. This implies for a limit in $q$ variables,

\begin{align*} \lim _{u_1,\ldots,u_q \rightarrow a} \frac{1}{\Delta (u_q,\ldots,u_1)}\det \big (f_j(u_i)\big )_{i,j=1}^J = \lim _{u_q\rightarrow a} \;\lim _{u_{q-1}\rightarrow u_q} \frac{1}{(u_q-q_{q-1})^{q-1}} \det \begin{pmatrix} \Big (\frac{1}{(i-1)!} f_j^{(i-1)}(u_{q-1})\Big )_{i \leq q-1, j \leq J} \\[15pt] \left(f_j(u_i)\right)_{i \geq q, j \leq J}\end{pmatrix}. \end{align*}

But Taylor expanding the entries of row $q$ as

\begin{equation*} f_j(u_q) = \sum _{i=1}^q \frac {f_j^{(i-1)}(u_{q-1})}{(i-1)!} \left(u_q-u_{q-1}\right)^{i-1} + O\left(\left(u_q-u_{q-1}\right)^{q}\right), \end{equation*}

and using multilinearity of the determinant to cancel out the first $q-1$ terms of the above sum in row $q$ , the claimed result quickly follows.

Remark 11. It is easy to see by permuting rows of the determinant that this result also implies

(3.2) \begin{equation} \lim _{u_1,\ldots,u_q \rightarrow a} \frac{1}{\Delta \left(u_1,\ldots,u_q\right)}\det \big (f_j(u_i)\big )_{i,j=1}^J = \det \begin{pmatrix} \Big (\frac{1}{(q-i)!} f_j^{(q-i)}(a)\Big )_{i \leq q, j \leq J} \\[15pt] \big (f_j(u_i)\big )_{i \geq q+1, j \leq J}\end{pmatrix} \end{equation}

and

(3.3) \begin{equation} \lim _{u_{q+1},\ldots,u_J \rightarrow a} \frac{1}{\Delta (u_J,\ldots,u_{q+1})}\det \big (f_j(u_i)\big )_{i,j=1}^J = \det \begin{pmatrix} \big (f_j(u_i)\big )_{i \leq q, j \leq J} \\[15pt] \Big (\frac{1}{(i - q-1)!} f_j^{(i - q-1)}(a)\Big )_{i \geq q+1, j \leq J}\end{pmatrix}, \end{equation}

where in this last equation the limit is taken in the order $u_{q+1}\rightarrow u_{q+2}$ ,…, $u_{J-1}\rightarrow u_J$ , $u_J \rightarrow a$ .

In applying this lemma, we need the following computation.

Lemma 12. For integers $\ell \geq 0$ and $N\geq 1$ ,

\begin{equation*} \lim _{u\rightarrow 0} \frac {1}{\ell !} \frac {d^\ell }{du^\ell } \left(\frac {1}{u-v} \frac {1-u^N v^N}{1-u^{2N}}\right) = -p_{N,\ell }(v), \end{equation*}

for $p_{N,\ell }$ defined by

(3.4) \begin{equation} p_{N,\ell }(v) \,:\!=\, \frac{1}{v^{\ell +1}} - v^{N-1} \, H_{N,\ell }(1/v) = \frac{1}{v^{\ell +1}} - \begin{cases} 0 & \textrm{if}\; 0 \leq [\ell ]_{2N} \leq N-1 \\[5pt] v^{2N-1-[\ell ]_{2N}} & \textrm{if}\; N \leq [\ell ]_{2N} \leq 2N-1.\end{cases} \end{equation}

Proof. Note that we have

\begin{align*} & \frac{1}{u-v} \frac{1-u^N v^N}{1-u^{2N}} = - \frac{1}{v} \frac{1}{1-u/v} + \frac{u^N-v^N}{u-v} \frac{u^N}{1-u^{2N}} \\[5pt]& = -\left(\frac{1}{v} + \frac{u}{v^2} + \frac{u^2}{v^3}+\cdots \right) + \left(v^{N-1} + v^{N-2} u + \cdots + u^{N-1}\right)\left(u^N + u^{3N} + \cdots \right), \end{align*}

taking a series expansion around $u=0$ . Since the quantity on the left-hand side of the Lemma is exactly the coefficient of $u^\ell$ in this expansion, the claim follows by inspection.

Lemma 13. (Cauchy Determinant Formula) For $u_1,\ldots,u_J$ and $v_1,\ldots,v_J$ collections of complex numbers with no elements in common,

\begin{equation*} \det \left(\frac {1}{u_i - v_j}\right)_{i,j=1}^J = \frac {\Delta (u_J,\ldots,u_1) \Delta (v_1,\ldots,v_J)}{\square (u;v)} \end{equation*}

where

\begin{equation*} \square (u;v)\,:\!=\, \prod _{i=1}^J \prod _{j=1}^J (u_i-v_j). \end{equation*}

Proof. See for instance [Reference Pólya and Szegő26, Part 7, Section 1, Ex. 3].

We can now give a proof of the moment formula for ACUE.

Proof of Theorem 2. We set $u = (u_1,\ldots,u_K)$ and $u^{\prime} = (u^{\prime}_1,\ldots,u^{\prime}_L)$ as abbreviations for ordered lists, and let $u \cup u^{\prime}\,:\!=\, (u_1,\ldots,u_K, u^{\prime}_1,\ldots,u^{\prime}_L)$ be an (ordered) concatenation of these lists. We abbreviate $\Delta (u) = \Delta (u_1,\ldots,u_K)$ and also use the notation $\widetilde{\Delta }(u) = \Delta (u_K,\ldots,u_1) = ({-}1)^{K(K-1)/2}\Delta (u)$ .

Our starting point is the identity

(3.5) \begin{align} \mathbb{E}_{\textrm{ACUE}(N)}\left[\det (g)^{-K} \prod _{k=1}^{K+L}\det (1+v_k g)\right] = \lim _{u \rightarrow \infty } u_1^N \cdots u_K^N \lim _{u^{\prime} \rightarrow 0} E_N(u \cup u^{\prime};\, v), \end{align}

where we define

\begin{equation*} E_N(u \cup u^{\prime};\, v) \,:\!=\, \mathbb {E}_{\textrm {ACUE}(N)}\Bigg [\frac {\prod _{k=1}^{K+L} \det (1+v_k g)}{\prod _{k=1}^K \det (1+u_k g) \prod _{\ell =1}^L \det (1+ u_\ell ^{\prime} g)}\Bigg ]. \end{equation*}

The limits $u \rightarrow \infty$ and $u^{\prime} \rightarrow 0$ mean $u_1,\ldots,u_K \rightarrow \infty$ and $u^{\prime}_1,\ldots,u^{\prime}_L \rightarrow 0$ . In what follows we will take these in the order $u^{\prime}_1 \rightarrow u^{\prime}_2$ , …, $u^{\prime}_{L-1}\rightarrow u^{\prime}_L$ , $u^{\prime}_L\rightarrow 0$ and $u_1 \rightarrow u_2$ , …, $u_{K-1}\rightarrow u_K$ , $u_K\rightarrow \infty$ so that Lemma 9 can easily be applied.

For notational reasons, we write

\begin{equation*} f_N(u,v)\,:\!=\, \frac {1}{u-v}\frac {1-u^N v^N}{1-u^{2N}}. \end{equation*}

We use Theorem 4 and Lemma 13 to see,

\begin{align*} E_N(u \cup u^{\prime};\, v) &= \frac{\square (u \cup u^{\prime};\, v)}{\widetilde{\Delta }(u\cup u^{\prime}) \Delta (v)} \det \begin{pmatrix} \Big (f_N\left(u_i,v_j\right)\Big )_{i\leq K, j \leq K+L} \\[15pt] \Big (f_N(u^{\prime}_{i-K},v_j)\Big )_{i\geq K+1, j \leq K+L} \end{pmatrix} \\[5pt] &= \frac{\square (u ;\, v) \square (u^{\prime} ;\, v)}{\widetilde{\Delta }(u) \widetilde{\Delta }(u^{\prime}) \square (u^{\prime};u) \Delta (v)} \det \begin{pmatrix} \Big (f_N\left(u_i,v_j\right)\Big )_{i\leq K, j \leq K+L} \\[15pt] \Big (f_N(u^{\prime}_{i-K},v_j)\Big )_{i\geq K+1, j \leq K+L} \end{pmatrix}. \end{align*}

Taking a limit $u^{\prime}\rightarrow 0$ and using Lemma 9—in particular its consequence (3.3)—and Lemma 12,

\begin{equation*} \lim _{u^{\prime}\rightarrow 0} E_N(u \cup u^{\prime};\, v) = \frac {\square (u;v) \prod _{k=1}^{K+L}({-}v_k)^L}{\widetilde {\Delta }(u) \prod _{k=1}^K ({-}u_k)^L \Delta (v)} \det \begin {pmatrix} \Big (f_N\left(u_i,v_j\right)\Big )_{i\leq K, j \leq K+L} \\[15pt] \Big ({-}p_{N,i-K-1}(v_j)\Big )_{i\geq K+1, j \leq K+L} \end {pmatrix}. \end{equation*}

But note the easily verified functional equation

\begin{equation*} f_N(u,v) = -f_N\left(u^{-1}, v^{-1}\right) v^{N-1} u^{-(N+1)}. \end{equation*}

Thus,

(3.6) \begin{align} u_1^N\cdots u_K^N \lim _{u^{\prime}\rightarrow 0} E_N(u \cup u^{\prime};\, v) = ({-}1)^L \frac{\square (u;v) \prod _{k=1}^{K+L} v_k^L}{\widetilde{\Delta }(u) \Delta (v)} \prod _{k=1}^K u_k^{-L-1} \cdot \det \begin{pmatrix} \left({-} v_j^{N-1}f_N\left(u_i^{-1},v_j^{-1}\right)\right)_{i\leq K, j \leq K+L} \\[15pt] \Big ({-}p_{N,i-K-1}(v_j)\Big )_{i\geq K+1, j \leq K+L} \end{pmatrix}. \end{align}

For fixed $v$ , we have as $u\rightarrow \infty$ ,

\begin{equation*} \frac {\square (u;v)}{\widetilde {\Delta }(u)} \prod _{k=1}^K u_k^{-L-1} = \frac {\square (u;v) \prod _{k=1}^K u_k^{-L-1}}{\prod _{k=1}^K u_k^{K-1} \Delta \left(\frac {1}{u_1},\ldots,\frac {1}{u_K}\right)} \sim \frac {1}{\Delta \left(\frac {1}{u_1},\ldots,\frac {1}{u_K}\right)}. \end{equation*}

Applying Lemma (2.4)—with its consequence (3.2) this time—and Lemma 12, the limit of (3.6) as $u \rightarrow \infty$ is

\begin{align*} & = ({-}1)^L \frac{\prod _{k=1}^{K+L} v_k^L}{\Delta (v)} \det \begin{pmatrix} \Big ( v_j^{N-1}p_{N,K-i}(v_j^{-1})\Big )_{i\leq K, j \leq K+L} \\[15pt] \Big ({-}p_{N,i-K-1}(v_j)\Big )_{i\geq K+1, j \leq K+L} \end{pmatrix} \\[5pt] & = \frac{1}{\Delta (v)} \det \begin{pmatrix} \Big ( v_j^{N+L-1}p_{N,K-i}(v_j^{-1})\Big )_{i\leq K, j \leq K+L} \\[15pt] \Big ( v_j^L p_{N,i-K-1}(v_j)\Big )_{i\geq K+1, j \leq K+L} \end{pmatrix}. \end{align*}

By inspection of matrix entries, the above is

\begin{equation*} = \frac {\det \big (\phi _i(v_j)\big )_{i,j=1}^{K+L}}{\Delta (v)}. \end{equation*}

Recalling (3.5), this is exactly what we sought to prove.

4. Hypothetical implications for ratios of $\zeta (s)$

Let us briefly and somewhat informally discuss these results in the context of the distribution of the Riemann zeta-function. For the sake of this discussion, suppose the Riemann Hypothesis is true, and label the nontrivial zeros of the zeta-function by $\{1/2+i\gamma _j\}_{j\in \mathbb{Z}}$ , so that $\gamma _j \in \mathbb{R}$ for all $j$ . What is widely believed about the local distribution of zeros concerns two point processes, the first point process (associated with a large parameter $T$ ) given by

(4.1) \begin{equation} \left\{ \frac{\log T}{2\pi }(\gamma _j-t)\right\}_{j \in \mathbb{Z}} \end{equation}

where $t \in [T,2T]$ is chosen randomly and uniformly, and the second point process (associated with a large parameter $N$ ) given by

(4.2) \begin{equation} \{N\theta _i\}_{i=1,\ldots,N} \end{equation}

where $\theta _1,\ldots,\theta _N \in [-1/2,1/2)$ are identified with the points $e(\theta _1),\ldots,e(\theta _N)$ of $\textrm{CUE}(N)$ . The widely believed GUE Hypothesis states that as $T\rightarrow \infty$ and $N\rightarrow \infty$ both point processes (4.1) and (4.2) tend to have the same limiting point process. (This means that randomly generated configurations of points from these two processes will look similar near the origin of the real line.)

The name GUE Hypothesis has historical origins; GUE refers to the Gaussian Unitary Ensemble, an ensemble of random matrices which, like CUE, locally tends to this same limiting point process, but which was investigated earlier. Much of what is known rigorously about the limiting distribution of the points in (4.1) is due to Montgomery [Reference Montgomery24], Hejhal [Reference Hejhal16], and Rudnick and Sarnak [Reference Rudnick and Sarnak28], who showed that the correlation functions of the points in (4.1) agree with those of this limiting distribution up to a band-limited resolution. Numerical work, beginning with that of Odlyzko [Reference Odlyzko25], has given further support to the GUE Hypothesis.

The ACUE was first investigated as one alternative model of how zeros of the Riemann zeta-function might be spaced. In particular, one considers the point process (associated with a large parameter $N$ ) given by

(4.3) \begin{equation} \left\{N\vartheta _i + \tfrac{r}{2}\right\}_{i=1,\ldots,N} \end{equation}

where $\vartheta _1,\ldots,\vartheta _N \in \left\{-\tfrac{1}{2}, \tfrac{-N+1}{2N},\tfrac{-N+2}{2N},\ldots, \tfrac{N-1}{2N}\right\}$ are identified with the points $e(\vartheta _1),\ldots,e(\vartheta _N)$ of $\textrm{ACUE}(N)$ , and $r \in [0,1)$ is chosen independently, and uniformly at random. As $N\rightarrow \infty$ , the point process (4.3) tends to a limiting process, called the AH point process in [Reference Lagarias and Rodgers22]. The AH point process has correlation functions which mimic the limiting process for CUE in a fashion akin to what is known rigorously about zeta zeros from the results of Montgomery, Hejhal, and Rudnick and Sarnak (see [Reference Lagarias and Rodgers21] for further discussion), but it also has gaps between points which are always half-integers. In this way, it is one possible—though likely not a unique—candidate for a limiting distribution of the zeta-function point process (4.1) which is compatible with what is currently known about the local distribution of zeros of the zeta-function and also with the so-called Alternative Hypothesis, a (widely disbelieved) conjecture that gaps between zeros always occur close to half-integer multiples of the mean spacing.

For this reason, [Reference Tao32] gave the name AGUE (Alternative Gaussian Unitary Ensemble) Hypothesis to the hypothetical claim that as $T\rightarrow \infty$ the zeta zero point process (4.1) tends to the AH point process. As one would like to rule out the Alternative Hypothesis, one would like to rule out the stronger AGUE Hypothesis.

More details on the AH point process can be found in the references [Reference Lagarias and Rodgers22, Reference Tao32], while further information on the Alternative Hypothesis in general can be found in [Reference Baluyot2].

A major impetus for studying mixed moments of characteristic polynomials $\det (1+u G)$ for the CUE came from the work of Keating-Snaith, who used information about CUE moments to make a conjecture regarding moments of the Riemann zeta-function [Reference Keating and Snaith19, Eq. (19)]. As first observed by Tao and as discussed in the introduction, the consequence of Theorem 1 that for sufficiently large $N$ mixed moments in the CUE and ACUE agree suggests that even should the zeros of the Riemann zeta-function be spaced according to the pattern of the ACUE, this could still be consistent with the Keating-Snaith moment conjecture.

The local spacing of zeros of the Riemann zeta-function is also closely related to the averages of ratios of shifts of the Riemann zeta-function near the critical line. This perspective was first pursued by Farmer [Reference Farmer14, Reference Farmer15] and has subsequently been investigated by others [Reference Conrey, Farmer and Zirnbauer10, Reference Conrey and Snaith12, Reference Conrey and Snaith13]. In particular, note that from Theorem 5,

\begin{equation*} \lim _{N\rightarrow \infty }\mathbb {E}_{\textrm {CUE}(N)} \Bigg [\prod _{j=1}^J\frac {\det (1 - e^{-\nu _j/N} G)}{\det (1 - e^{-\mu _j/N} G)} \Bigg ] = \frac {1}{\det \Big (\frac {1}{\nu _j - \mu _i}\Big )} \det \left (\frac {1}{\nu _j-\mu _i} e\left(\mu _i,\nu _j\right)\right ), \end{equation*}

for $\operatorname{Re}\, \mu _j \neq 0$ for all $j$ , where

\begin{equation*} e(\mu,\nu ) \,:\!=\, \begin {cases} 1 & \textrm {if}\; \operatorname {Re}\, \mu \gt 0 \\ e^{\mu -\nu } & \textrm {if}\; \operatorname {Re}\, \mu \lt 0.\end {cases} \end{equation*}

From the results proved in [Reference Conrey and Snaith13, Reference Rodgers27], it can be seen that the claim

(4.4) \begin{equation} \lim _{T\rightarrow \infty }\frac{1}{T}\int _T^{2T} \prod _{j=1}^J \frac{\zeta (1/2 + \nu _j/\log T + it)}{\zeta (1/2 + \mu _j/\log T + it)} \, dt = \frac{1}{\det \Big (\frac{1}{\nu _j - \mu _i}\Big )} \det \left (\frac{1}{\nu _j-\mu _i} e\left(\mu _i,\nu _j\right)\right ) \end{equation}

for $\operatorname{Re}\, \mu _j \neq 0$ for all $j$ , is equivalent to the GUE Hypothesis. (In fact [Reference Rodgers27] treats only real $\mu _j, \nu _j$ , but the method can be adapted to complex values. There is a notational difference in [Reference Rodgers27]; the function $E$ used there satisfies $E(\nu,\mu ) = e(\mu,\nu )$ for the function $e$ used here.)

A belief in the AGUE Hypothesis would suggest that we replace characteristic polynomials $\det (1-u G)$ as they appear above by $\det (1- u e^{i 2\pi r/2N} g)$ , where $r\in [0,1)$ is independent of $g$ and uniformly chosen. For the ACUE, from Theorem 4 we have

\begin{equation*} \lim _{N\rightarrow \infty } \mathbb {E}_{\textrm {ACUE}(N)} \Bigg [\prod _{j=1}^J\frac {\det (1 - e^{-\nu _j/N} g)}{\det (1 - e^{-\mu _j/N} g)} \Bigg ] = \frac {1}{\det \Big (\frac {1}{\nu _j - \mu _i}\Big )} \det \left (\frac {1}{\nu _j-\mu _i} \mathfrak {e}\left(\mu _i,\nu _j\right)\right ), \end{equation*}

for $\mu _j \notin \frac{i}{2}\mathbb{Z}$ for all $j$ , where

\begin{equation*} \mathfrak {e}(\mu,\nu ) \,:\!=\, \frac {1 - e^{-\mu -\nu }}{1 - e^{-2\mu }}. \end{equation*}

Hence on the assumption of the AGUE Hypothesis, one should instead expect for $\operatorname{Re} \, \mu _j \neq 0$ for all $j$ ,

(4.5) \begin{align} & \lim _{T\rightarrow \infty }\frac{1}{T}\int _T^{2T} \prod _{j=1}^J \frac{\zeta (1/2 + \nu _j/\log T + it)}{\zeta (1/2 + \mu _j/\log T + it)}\, dt \nonumber\\& = \lim _{N\rightarrow \infty } \int _0^1 \mathbb{E}_{\textrm{ACUE}(N)} \Bigg [\prod _{j=1}^J\frac{\det (1 - e^{-\nu _j/N}e^{i\pi r/N} g)}{\det (1 - e^{-\mu _j/N} e^{i\pi r/N} g)} \Bigg ]\, dr\nonumber\\ & = \int _0^1 \frac{1}{\det \Big (\frac{1}{\nu _j - \mu _i}\Big )} \det \left (\frac{1}{\nu _j-\mu _i} \mathfrak{e}(\mu _i-i\pi r,\nu _j-i\pi r)\right ) dr \nonumber\\ & = \frac{1}{\det \Big (\frac{1}{\nu _j - \mu _i}\Big )}\int _{|z|=1} \det \left (\frac{1}{\nu _j-\mu _i} \frac{1 - z e^{-\mu _i-\nu _j}}{1 - z e^{-2\mu _i}}\right ) \frac{dz}{z}. \end{align}

(4.5) is of course a different expression than (4.4). Thus for averages of ratios of the Riemann zeta-function, an ACUE spacing would be distinguished from CUE spacing. In fact using the methods of [Reference Conrey and Snaith13, Reference Rodgers27], it should be possible to demonstrate rigorously that (4.5) is equivalent to the AGUE Hypothesis, but we do not pursue this here.