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This is another sequel to a recent post in which I showed the Riemann zeta function ${\zeta}$ can be locally approximated by a polynomial, in the sense that for randomly chosen ${t \in [T,2T]}$ one has an approximation

$\displaystyle \zeta(\frac{1}{2} + it - \frac{2\pi i z}{\log T}) \approx P_t( e^{2\pi i z/N} ) \ \ \ \ \ (1)$

where ${N}$ grows slowly with ${T}$, and ${P_t}$ is a polynomial of degree ${N}$. It turns out that in the function field setting there is an exact version of this approximation which captures many of the known features of the Riemann zeta function, namely Dirichlet ${L}$-functions for a random character of given modulus over a function field. This model was (essentially) studied in a fairly recent paper by Andrade, Miller, Pratt, and Trinh; I am not sure if there is any further literature on this model beyond this paper (though the number field analogue of low-lying zeroes of Dirichlet ${L}$-functions is certainly well studied). In this model it is possible to set ${N}$ fixed and let ${T}$ go to infinity, thus providing a simple finite-dimensional model problem for problems involving the statistics of zeroes of the zeta function.

In this post I would like to record this analogue precisely. We will need a finite field ${{\mathbb F}}$ of some order ${q}$ and a natural number ${N}$, and set

$\displaystyle T := q^{N+1}.$

We will primarily think of ${q}$ as being large and ${N}$ as being either fixed or growing very slowly with ${q}$, though it is possible to also consider other asymptotic regimes (such as holding ${q}$ fixed and letting ${N}$ go to infinity). Let ${{\mathbb F}[X]}$ be the ring of polynomials of one variable ${X}$ with coefficients in ${{\mathbb F}}$, and let ${{\mathbb F}[X]'}$ be the multiplicative semigroup of monic polynomials in ${{\mathbb F}[X]}$; one should view ${{\mathbb F}[X]}$ and ${{\mathbb F}[X]'}$ as the function field analogue of the integers and natural numbers respectively. We use the valuation ${|n| := q^{\mathrm{deg}(n)}}$ for polynomials ${n \in {\mathbb F}[X]}$ (with ${|0|=0}$); this is the analogue of the usual absolute value on the integers. We select an irreducible polynomial ${Q \in {\mathbb F}[X]}$ of size ${|Q|=T}$ (i.e., ${Q}$ has degree ${N+1}$). The multiplicative group ${({\mathbb F}[X]/Q{\mathbb F}[X])^\times}$ can be shown to be cyclic of order ${|Q|-1=T-1}$. A Dirichlet character of modulus ${Q}$ is a completely multiplicative function ${\chi: {\mathbb F}[X] \rightarrow {\bf C}}$ of modulus ${Q}$, that is periodic of period ${Q}$ and vanishes on those ${n \in {\mathbb F}[X]}$ not coprime to ${Q}$. From Fourier analysis we see that there are exactly ${\phi(Q) := |Q|-1}$ Dirichlet characters of modulus ${Q}$. A Dirichlet character is said to be odd if it is not identically one on the group ${{\mathbb F}^\times}$ of non-zero constants; there are only ${\frac{1}{q-1} \phi(Q)}$ non-odd characters (including the principal character), so in the limit ${q \rightarrow \infty}$ most Dirichlet characters are odd. We will work primarily with odd characters in order to be able to ignore the effect of the place at infinity.

Let ${\chi}$ be an odd Dirichlet character of modulus ${Q}$. The Dirichlet ${L}$-function ${L(s, \chi)}$ is then defined (for ${s \in {\bf C}}$ of sufficiently large real part, at least) as

$\displaystyle L(s,\chi) := \sum_{n \in {\mathbb F}[X]'} \frac{\chi(n)}{|n|^s}$

$\displaystyle = \sum_{m=0}^\infty q^{-sm} \sum_{n \in {\mathbb F}[X]': |n| = q^m} \chi(n).$

Note that for ${m \geq N+1}$, the set ${n \in {\mathbb F}[X]': |n| = q^m}$ is invariant under shifts ${h}$ whenever ${|h| < T}$; since this covers a full set of residue classes of ${{\mathbb F}[X]/Q{\mathbb F}[X]}$, and the odd character ${\chi}$ has mean zero on this set of residue classes, we conclude that the sum ${\sum_{n \in {\mathbb F}[X]': |n| = q^m} \chi(n)}$ vanishes for ${m \geq N+1}$. In particular, the ${L}$-function is entire, and for any real number ${t}$ and complex number ${z}$, we can write the ${L}$-function as a polynomial

$\displaystyle L(\frac{1}{2} + it - \frac{2\pi i z}{\log T},\chi) = P(Z) = P_{t,\chi}(Z) := \sum_{m=0}^N c^1_m(t,\chi) Z^j$

where ${Z := e(z/N) = e^{2\pi i z/N}}$ and the coefficients ${c^1_m = c^1_m(t,\chi)}$ are given by the formula

$\displaystyle c^1_m(t,\chi) := q^{-m/2-imt} \sum_{n \in {\mathbb F}[X]': |n| = q^m} \chi(n).$

Note that ${t}$ can easily be normalised to zero by the relation

$\displaystyle P_{t,\chi}(Z) = P_{0,\chi}( q^{-it} Z ). \ \ \ \ \ (2)$

In particular, the dependence on ${t}$ is periodic with period ${\frac{2\pi}{\log q}}$ (so by abuse of notation one could also take ${t}$ to be an element of ${{\bf R}/\frac{2\pi}{\log q}{\bf Z}}$).

Fourier inversion yields a functional equation for the polynomial ${P}$:

Proposition 1 (Functional equation) Let ${\chi}$ be an odd Dirichlet character of modulus ${Q}$, and ${t \in {\bf R}}$. There exists a phase ${e(\theta)}$ (depending on ${t,\chi}$) such that

$\displaystyle a_{N-m}^1 = e(\theta) \overline{c^1_m}$

for all ${0 \leq m \leq N}$, or equivalently that

$\displaystyle P(1/Z) = e^{i\theta} Z^{-N} \overline{P}(Z)$

where ${\overline{P}(Z) := \overline{P(\overline{Z})}}$.

Proof: We can normalise ${t=0}$. Let ${G}$ be the finite field ${{\mathbb F}[X] / Q {\mathbb F}[X]}$. We can write

$\displaystyle a_{N-m} = q^{-(N-m)/2} \sum_{n \in q^{N-m} + H_{N-m}} \chi(n)$

where ${H_j}$ denotes the subgroup of ${G}$ consisting of (residue classes of) polynomials of degree less than ${j}$. Let ${e_G: G \rightarrow S^1}$ be a non-trivial character of ${G}$ whose kernel lies in the space ${H_N}$ (this is easily achieved by pulling back a non-trivial character from the quotient ${G/H_N \equiv {\mathbb F}}$). We can use the Fourier inversion formula to write

$\displaystyle a_{N-m} = q^{(m-N)/2} \sum_{\xi \in G} \hat \chi(\xi) \sum_{n \in T^{N-m} + H_{N-m}} e_G( n\xi )$

where

$\displaystyle \hat \chi(\xi) := q^{-N-1} \sum_{n \in G} \chi(n) e_G(-n\xi).$

From change of variables we see that ${\hat \chi}$ is a scalar multiple of ${\overline{\chi}}$; from Plancherel we conclude that

$\displaystyle \hat \chi = e(\theta_0) q^{-(N+1)/2} \overline{\chi} \ \ \ \ \ (3)$

for some phase ${e(\theta_0)}$. We conclude that

$\displaystyle a_{N-m} = e(\theta_0) q^{-(2N-m+1)/2} \sum_{\xi \in G} \overline{\chi}(\xi) e_G( T^{N-j} \xi) \sum_{n \in H_{N-j}} e_G( n\xi ). \ \ \ \ \ (4)$

The inner sum ${\sum_{n \in H_{N-m}} e_G( n\xi )}$ equals ${q^{N-m}}$ if ${\xi \in H_{j+1}}$, and vanishes otherwise, thus

$\displaystyle a_{N-m} = e(\theta_0) q^{-(m+1)/2} \sum_{\xi \in H_{j+1}} \overline{\chi}(\xi) e_G( T^{N-m} \xi).$

For ${\xi}$ in ${H_j}$, ${e_G(T^{N-m} \xi)=1}$ and the contribution of the sum vanishes as ${\chi}$ is odd. Thus we may restrict ${\xi}$ to ${H_{m+1} \backslash H_m}$, so that

$\displaystyle a_{N-m} = e(\theta_0) q^{-(m+1)/2} \sum_{h \in {\mathbb F}^\times} e_G( T^{N} h) \sum_{\xi \in h T^m + H_{m}} \overline{\chi}(\xi).$

By the multiplicativity of ${\chi}$, this factorises as

$\displaystyle a_{N-m} = e(\theta_0) q^{-(m+1)/2} (\sum_{h \in {\mathbb F}^\times} \overline{\chi}(h) e_G( T^{N} h)) (\sum_{\xi \in T^m + H_{m}} \overline{\chi}(\xi)).$

From the one-dimensional version of (3) (and the fact that ${\chi}$ is odd) we have

$\displaystyle \sum_{h \in {\mathbb F}^\times} \overline{\chi}(h) e_G( T^{N} h) = e(\theta_1) q^{1/2}$

for some phase ${e(\theta_1)}$. The claim follows. $\Box$

As one corollary of the functional equation, ${a_N}$ is a phase rotation of ${\overline{a_1} = 1}$ and thus is non-zero, so ${P}$ has degree exactly ${N}$. The functional equation is then equivalent to the ${N}$ zeroes of ${P}$ being symmetric across the unit circle. In fact we have the stronger

Theorem 2 (Riemann hypothesis for Dirichlet ${L}$-functions over function fields) Let ${\chi}$ be an odd Dirichlet character of modulus ${Q}$, and ${t \in {\bf R}}$. Then all the zeroes of ${P}$ lie on the unit circle.

We derive this result from the Riemann hypothesis for curves over function fields below the fold.

In view of this theorem (and the fact that ${a_1=1}$), we may write

$\displaystyle P(Z) = \mathrm{det}(1 - ZU)$

for some unitary ${N \times N}$ matrix ${U = U_{t,\chi}}$. It is possible to interpret ${U}$ as the action of the geometric Frobenius map on a certain cohomology group, but we will not do so here. The situation here is simpler than in the number field case because the factor ${\exp(A)}$ arising from very small primes is now absent (in the function field setting there are no primes of size between ${1}$ and ${q}$).

We now let ${\chi}$ vary uniformly at random over all odd characters of modulus ${Q}$, and ${t}$ uniformly over ${{\bf R}/\frac{2\pi}{\log q}{\bf Z}}$, independently of ${\chi}$; we also make the distribution of the random variable ${U}$ conjugation invariant in ${U(N)}$. We use ${{\mathbf E}_Q}$ to denote the expectation with respect to this randomness. One can then ask what the limiting distribution of ${U}$ is in various regimes; we will focus in this post on the regime where ${N}$ is fixed and ${q}$ is being sent to infinity. In the spirit of the Sato-Tate conjecture, one should expect ${U}$ to converge in distribution to the circular unitary ensemble (CUE), that is to say Haar probability measure on ${U(N)}$. This may well be provable from Deligne’s “Weil II” machinery (in the spirit of this monograph of Katz and Sarnak), though I do not know how feasible this is or whether it has already been done in the literature; here we shall avoid using this machinery and study what partial results towards this CUE hypothesis one can make without it.

If one lets ${\lambda_1,\dots,\lambda_N}$ be the eigenvalues of ${U}$ (ordered arbitrarily), then we now have

$\displaystyle \sum_{m=0}^N c^1_m Z^m = P(Z) = \prod_{j=1}^N (1 - \lambda_j Z)$

and hence the ${c^1_m}$ are essentially elementary symmetric polynomials of the eigenvalues:

$\displaystyle c^1_m = (-1)^j e_m( \lambda_1,\dots,\lambda_N). \ \ \ \ \ (5)$

One can take log derivatives to conclude

$\displaystyle \frac{P'(Z)}{P(Z)} = \sum_{j=1}^N \frac{\lambda_j}{1-\lambda_j Z}.$

On the other hand, as in the number field case one has the Dirichlet series expansion

$\displaystyle Z \frac{P'(Z)}{P(Z)} = \sum_{n \in {\mathbb F}[X]'} \frac{\Lambda_q(n) \chi(n)}{|n|^s}$

where ${s = \frac{1}{2} + it - \frac{2\pi i z}{\log T}}$ has sufficiently large real part, ${Z = e(z/N)}$, and the von Mangoldt function ${\Lambda_q(n)}$ is defined as ${\log_q |p| = \mathrm{deg} p}$ when ${n}$ is the power of an irreducible ${p}$ and ${0}$ otherwise. We conclude the “explicit formula”

$\displaystyle c^{\Lambda_q}_m = \sum_{j=1}^N \lambda_j^m = \mathrm{tr}(U^m) \ \ \ \ \ (6)$

for ${m \geq 1}$, where

$\displaystyle c^{\Lambda_q}_m := q^{-m/2-imt} \sum_{n \in {\mathbb F}[X]': |n| = q^m} \Lambda_q(n) \chi(n).$

Similarly on inverting ${P(Z)}$ we have

$\displaystyle P(Z)^{-1} = \prod_{j=1}^N (1 - \lambda_j Z)^{-1}.$

Since we also have

$\displaystyle P(Z)^{-1} = \sum_{n \in {\mathbb F}[X]'} \frac{\mu(n) \chi(n)}{|n|^s}$

for ${s}$ sufficiently large real part, where the Möbius function ${\mu(n)}$ is equal to ${(-1)^k}$ when ${n}$ is the product of ${k}$ distinct irreducibles, and ${0}$ otherwise, we conclude that the Möbius coefficients

$\displaystyle c^\mu_m := q^{-m/2-imt} \sum_{n \in {\mathbb F}[X]': |n| = q^m} \mu(n) \chi(n)$

are just the complete homogeneous symmetric polynomials of the eigenvalues:

$\displaystyle c^\mu_m = h_m( \lambda_1,\dots,\lambda_N). \ \ \ \ \ (7)$

One can then derive various algebraic relationships between the coefficients ${c^1_m, c^{\Lambda_q}_m, c^\mu_m}$ from various identities involving symmetric polynomials, but we will not do so here.

What do we know about the distribution of ${U}$? By construction, it is conjugation-invariant; from (2) it is also invariant with respect to the rotations ${U \rightarrow e^{i\theta} U}$ for any phase ${\theta \in{\bf R}}$. We also have the function field analogue of the Rudnick-Sarnak asymptotics:

Proposition 3 (Rudnick-Sarnak asymptotics) Let ${a_1,\dots,a_k,b_1,\dots,b_k}$ be nonnegative integers. If

$\displaystyle \sum_{j=1}^k j a_j \leq N, \ \ \ \ \ (8)$

then the moment

$\displaystyle {\bf E}_{Q} \prod_{j=1}^k (\mathrm{tr} U^j)^{a_j} (\overline{\mathrm{tr} U^j})^{b_j} \ \ \ \ \ (9)$

is equal to ${o(1)}$ in the limit ${q \rightarrow \infty}$ (holding ${N,a_1,\dots,a_k,b_1,\dots,b_k}$ fixed) unless ${a_j=b_j}$ for all ${j}$, in which case it is equal to

$\displaystyle \prod_{j=1}^k j^{a_j} a_j! + o(1). \ \ \ \ \ (10)$

Comparing this with Proposition 1 from this previous post, we thus see that all the low moments of ${U}$ are consistent with the CUE hypothesis (and also with the ACUE hypothesis, again by the previous post). The case ${\sum_{j=1}^k a_j + \sum_{j=1}^k b_j \leq 2}$ of this proposition was essentially established by Andrade, Miller, Pratt, and Trinh.

Proof: We may assume the homogeneity relationship

$\displaystyle \sum_{j=1}^k j a_j = \sum_{j=1}^k j b_j \ \ \ \ \ (11)$

since otherwise the claim follows from the invariance under phase rotation ${U \mapsto e^{i\theta} U}$. By (6), the expression (9) is equal to

$\displaystyle q^{-D} {\bf E}_Q \sum_{n_1,\dots,n_l,n'_1,\dots,n'_{l'} \in {\mathbb F}[X]': |n_i| = q^{s_i}, |n'_i| = q^{s'_i}} (\prod_{i=1}^l \Lambda_q(n_i) \chi(n_i)) \prod_{i=1}^{l'} \Lambda_q(n'_i) \overline{\chi(n'_i)}$

where

$\displaystyle D := \sum_{j=1}^k j a_j = \sum_{j=1}^k j b_j$

$\displaystyle l := \sum_{j=1}^k a_j$

$\displaystyle l' := \sum_{j=1}^k b_j$

and ${s_1 \leq \dots \leq s_l}$ consists of ${a_j}$ copies of ${j}$ for each ${j=1,\dots,k}$, and similarly ${s'_1 \leq \dots \leq s'_{l'}}$ consists of ${b_j}$ copies of ${j}$ for each ${j=1,\dots,k}$.

The polynomials ${n_1 \dots n_l}$ and ${n'_1 \dots n'_{l'}}$ are monic of degree ${D}$, which by hypothesis is less than the degree of ${Q}$, and thus they can only be scalar multiples of each other in ${{\mathbb F}[X] / Q {\mathbb F}[X]}$ if they are identical (in ${{\mathbb F}[X]}$). As such, we see that the average

$\displaystyle {\bf E}_Q \chi(n_1) \dots \chi(n_l) \overline{\chi(n'_1)} \dots \overline{\chi(n'_{l'})}$

vanishes unless ${n_1 \dots n_l = n'_1 \dots n'_{l'}}$, in which case this average is equal to ${1}$. Thus the expression (9) simplifies to

$\displaystyle q^{-D} \sum_{n_1,\dots,n_l,n'_1,\dots,n'_{l'}: |n_i| = q^{s_i}, |n'_i| = q^{s'_i}; n_1 \dots n_l = n'_1 \dots n'_l} (\prod_{i=1}^l \Lambda_q(n_i)) \prod_{i=1}^{l'} \Lambda_q(n'_i).$

There are at most ${q^D}$ choices for the product ${n_1 \dots n_l}$, and each one contributes ${O_D(1)}$ to the above sum. All but ${o(q^D)}$ of these choices are square-free, so by accepting an error of ${o(1)}$, we may restrict attention to square-free ${n_1 \dots n_l}$. This forces ${n_1,\dots,n_l,n'_1,\dots,n'_{l'}}$ to all be irreducible (as opposed to powers of irreducibles); as ${{\mathbb F}[X]}$ is a unique factorisation domain, this forces ${l=l'}$ and ${n_1,\dots,n_l}$ to be a permutation of ${n'_1,\dots,n'_{l'}}$. By the size restrictions, this then forces ${a_j = b_j}$ for all ${j}$ (if the above expression is to be anything other than ${o(1)}$), and each ${n_1,\dots,n_l}$ is associated to ${\prod_{j=1}^k a_j!}$ possible choices of ${n'_1,\dots,n'_{l'}}$. Writing ${\Lambda_q(n'_i) = s'_i}$ and then reinstating the non-squarefree possibilities for ${n_1 \dots n_l}$, we can thus write the above expression as

$\displaystyle q^{-D} \prod_{j=1}^k j a_j! \sum_{n_1,\dots,n_l,n'_1,\dots,n'_{l'}\in {\mathbb F}[X]': |n_i| = q^{s_i}} \prod_{i=1}^l \Lambda_q(n_i) + o(1).$

Using the prime number theorem ${\sum_{n \in {\mathbb F}[X]': |n| = q^s} \Lambda_q(n) = q^s}$, we obtain the claim. $\Box$

Comparing this with Proposition 1 from this previous post, we thus see that all the low moments of ${U}$ are consistent with the CUE and ACUE hypotheses:

Corollary 4 (CUE statistics at low frequencies) Let ${\lambda_1,\dots,\lambda_N}$ be the eigenvalues of ${U}$, permuted uniformly at random. Let ${R(\lambda)}$ be a linear combination of monomials ${\lambda_1^{a_1} \dots \lambda_N^{a_N}}$ where ${a_1,\dots,a_N}$ are integers with either ${\sum_{j=1}^N a_j \neq 0}$ or ${\sum_{j=1}^N |a_j| \leq 2N}$. Then

$\displaystyle {\bf E}_Q R(\lambda) = {\bf E}_{CUE} R(\lambda) + o(1).$

The analogue of the GUE hypothesis in this setting would be the CUE hypothesis, which asserts that the threshold ${2N}$ here can be replaced by an arbitrarily large quantity. As far as I know this is not known even for ${2N+2}$ (though, as mentioned previously, in principle one may be able to resolve such cases using Deligne’s proof of the Riemann hypothesis for function fields). Among other things, this would allow one to distinguish CUE from ACUE, since as discussed in the previous post, these two distributions agree when tested against monomials up to threshold ${2N}$, though not to ${2N+2}$.

Proof: By permutation symmetry we can take ${R}$ to be symmetric, and by linearity we may then take ${R}$ to be the symmetrisation of a single monomial ${\lambda_1^{a_1} \dots \lambda_N^{a_N}}$. If ${\sum_{j=1}^N a_j \neq 0}$ then both expectations vanish due to the phase rotation symmetry, so we may assume that ${\sum_{j=1}^N a_j \neq 0}$ and ${\sum_{j=1}^N |a_j| \leq 2N}$. We can write this symmetric polynomial as a constant multiple of ${\mathrm{tr}(U^{a_1}) \dots \mathrm{tr}(U^{a_N})}$ plus other monomials with a smaller value of ${\sum_{j=1}^N |a_j|}$. Since ${\mathrm{tr}(U^{-a}) = \overline{\mathrm{tr}(U^a)}}$, the claim now follows by induction from Proposition 3 and Proposition 1 from the previous post. $\Box$

Thus, for instance, for ${k=1,2}$, the ${2k^{th}}$ moment

$\displaystyle {\bf E}_Q |\det(1-U)|^{2k} = {\bf E}_Q |P(1)|^{2k} = {\bf E}_Q |L(\frac{1}{2} + it, \chi)|^{2k}$

is equal to

$\displaystyle {\bf E}_{CUE} |\det(1-U)|^{2k} + o(1)$

because all the monomials in ${\prod_{j=1}^N (1-\lambda_j)^k (1-\lambda_j^{-1})^k}$ are of the required form when ${k \leq 2}$. The latter expectation can be computed exactly (for any natural number ${k}$) using a formula

$\displaystyle {\bf E}_{CUE} |\det(1-U)|^{2k} = \prod_{j=1}^N \frac{\Gamma(j) \Gamma(j+2k)}{\Gamma(j+k)^2}$

of Baker-Forrester and Keating-Snaith, thus for instance

$\displaystyle {\bf E}_{CUE} |\det(1-U)|^2 = N+1$

$\displaystyle {\bf E}_{CUE} |\det(1-U)|^4 = \frac{(N+1)(N+2)^2(N+3)}{12}$

and more generally

$\displaystyle {\bf E}_{CUE}|\det(1-U)|^{2k} = \frac{g_k+o(1)}{(k^2)!} N^{k^2}$

when ${N \rightarrow \infty}$, where ${g_k}$ are the integers

$\displaystyle g_1 = 1, g_2 = 2, g_3 = 42, g_4 = 24024, \dots$

and more generally

$\displaystyle g_k := \frac{(k^2)!}{\prod_{i=1}^{2k-1} i^{k-|k-i|}}$

(OEIS A039622). Thus we have

$\displaystyle {\bf E}_Q |\det(1-U)|^{2k} = \frac{g_k+o(1)}{k^2!} N^{k^2}$

for ${k=1,2}$ if ${Q \rightarrow \infty}$ and ${N}$ is sufficiently slowly growing depending on ${Q}$. The CUE hypothesis would imply that that this formula also holds for higher ${k}$. (The situation here is cleaner than in the number field case, in which the GUE hypothesis only suggests the correct lower bound for the moments rather than an asymptotic, due to the absence of the wildly fluctuating additional factor ${\exp(A)}$ that is present in the Riemann zeta function model.)

Now we can recover the analogue of Montgomery’s work on the pair correlation conjecture. Consider the statistic

$\displaystyle {\bf E}_Q \sum_{1 \leq i,j \leq N} R( \lambda_i / \lambda_j )$

where

$\displaystyle R(z) = \sum_m \hat R(m) z^m$

is some finite linear combination of monomials ${z^m}$ independent of ${q}$. We can expand the above sum as

$\displaystyle \sum_m \hat R(m) {\bf E}_Q \mathrm{tr}(U^m) \mathrm{tr}(U^{-m}).$

Assuming the CUE hypothesis, then by Example 3 of the previous post, we would conclude that

$\displaystyle {\bf E}_Q \sum_{1 \leq i,j \leq N} R( \lambda_i / \lambda_j ) = N^2 \hat R(0) + \sum_m \min(|m|,N) \hat R(m) + o(1). \ \ \ \ \ (12)$

This is the analogue of Montgomery’s pair correlation conjecture. Proposition 3 implies that this claim is true whenever ${\hat R}$ is supported on ${[-N,N]}$. If instead we assume the ACUE hypothesis (or the weaker Alternative Hypothesis that the phase gaps are non-zero multiples of ${1/2N}$), one should instead have

$\displaystyle {\bf E}_Q \sum_{1 \leq i,j \leq N} R( \lambda_i / \lambda_j ) = \sum_{k \in {\bf Z}} N^2 \hat R(2Nk) + \sum_{1 \leq |m| \leq N} |m| \hat R(m+2Nk) + o(1)$

for arbitrary ${R}$; this is the function field analogue of a recent result of Baluyot. In any event, since ${\mathrm{tr}(U^m) \mathrm{tr}(U^{-m})}$ is non-negative, we unconditionally have the lower bound

$\displaystyle {\bf E}_Q \sum_{1 \leq i,j \leq N} R( \lambda_i / \lambda_j ) \geq N^2 \hat R(0) + \sum_{1 \leq |m| \leq N} |m| \hat R(m) + o(1). \ \ \ \ \ (13)$

if ${\hat R(m)}$ is non-negative for ${|m| > N}$.

By applying (12) for various choices of test functions ${R}$ we can obtain various bounds on the behaviour of eigenvalues. For instance suppose we take the Fejér kernel

$\displaystyle R(z) = |1 + z + \dots + z^N|^2 = \sum_{m=-N}^N (N+1-|m|) z^m.$

Then (12) applies unconditionally and we conclude that

$\displaystyle {\bf E}_Q \sum_{1 \leq i,j \leq N} R( \lambda_i / \lambda_j ) = N^2 (N+1) + \sum_{1 \leq |m| \leq N} (N+1-|m|) |m| + o(1).$

The right-hand side evaluates to ${\frac{2}{3} N(N+1)(2N+1)+o(1)}$. On the other hand, ${R(\lambda_i/\lambda_j)}$ is non-negative, and equal to ${(N+1)^2}$ when ${\lambda_i = \lambda_j}$. Thus

$\displaystyle {\bf E}_Q \sum_{1 \leq i,j \leq N} 1_{\lambda_i = \lambda_j} \leq \frac{2}{3} \frac{N(2N+1)}{N+1} + o(1).$

The sum ${\sum_{1 \leq j \leq N} 1_{\lambda_i = \lambda_j}}$ is at least ${1}$, and is at least ${2}$ if ${\lambda_i}$ is not a simple eigenvalue. Thus

$\displaystyle {\bf E}_Q \sum_{1 \leq i, \leq N} 1_{\lambda_i \hbox{ not simple}} \leq \frac{1}{3} \frac{N(N-1)}{N+1} + o(1),$

and thus the expected number of simple eigenvalues is at least ${\frac{2N}{3} \frac{N+4}{N+1} + o(1)}$; in particular, at least two thirds of the eigenvalues are simple asymptotically on average. If we had (12) without any restriction on the support of ${\hat R}$, the same arguments allow one to show that the expected proportion of simple eigenvalues is ${1-o(1)}$.

Suppose that the phase gaps in ${U}$ are all greater than ${c/N}$ almost surely. Let ${\hat R}$ is non-negative and ${R(e^{i\theta})}$ non-positive for ${\theta}$ outside of the arc ${[-c/N,c/N]}$. Then from (13) one has

$\displaystyle R(0) N \geq N^2 \hat R(0) + \sum_{1 \leq |m| \leq N} |m| \hat R(m) + o(1),$

so by taking contrapositives one can force the existence of a gap less than ${c/N}$ asymptotically if one can find ${R}$ with ${\hat R}$ non-negative, ${R}$ non-positive for ${\theta}$ outside of the arc ${[-c/N,c/N]}$, and for which one has the inequality

$\displaystyle R(0) N < N^2 \hat R(0) + \sum_{1 \leq |m| \leq N} |m| \hat R(m).$

By a suitable choice of ${R}$ (based on a minorant of Selberg) one can ensure this for ${c \approx 0.6072}$ for ${N}$ large; see Section 5 of these notes of Goldston. This is not the smallest value of ${c}$ currently obtainable in the literature for the number field case (which is currently ${0.50412}$, due to Goldston and Turnage-Butterbaugh, by a somewhat different method), but is still significantly less than the trivial value of ${1}$. On the other hand, due to the compatibility of the ACUE distribution with Proposition 3, it is not possible to lower ${c}$ below ${0.5}$ purely through the use of Proposition 3.

In some cases it is possible to go beyond Proposition 3. Consider the mollified moment

$\displaystyle {\bf E}_Q |M(U) P(1)|^2$

where

$\displaystyle M(U) = \sum_{m=0}^d a_m h_m(\lambda_1,\dots,\lambda_N)$

for some coefficients ${a_0,\dots,a_d}$. We can compute this moment in the CUE case:

Proposition 5 We have

$\displaystyle {\bf E}_{CUE} |M(U) P(1)|^2 = |a_0|^2 + N \sum_{m=1}^d |a_m - a_{m-1}|^2.$

Proof: From (5) one has

$\displaystyle P(1) = \sum_{i=0}^N (-1)^i e_i(\lambda_1,\dots,\lambda_N)$

hence

$\displaystyle M(U) P(1) = \sum_{i=0}^N \sum_{m=0}^d (-1)^i a_m e_i h_m$

where we suppress the dependence on the eigenvalues ${\lambda}$. Now observe the Pieri formula

$\displaystyle e_i h_m = s_{m 1^i} + s_{(m+1) 1^{i-1}}$

where ${s_{m 1^i}}$ are the hook Schur polynomials

$\displaystyle s_{m 1^i} = \sum_{a_1 \leq \dots \leq a_m; a_1 < b_1 < \dots < b_i} \lambda_{a_1} \dots \lambda_{a_m} \lambda_{b_1} \dots \lambda_{b_i}$

and we adopt the convention that ${s_{m 1^i}}$ vanishes for ${i = -1}$, or when ${m = 0}$ and ${i > 0}$. Then ${s_{m1^i}}$ also vanishes for ${i\geq N}$. We conclude that

$\displaystyle M(U) P(1) = a_0 s_{0 1^0} + \sum_{0 \leq i \leq N-1} \sum_{m \geq 1} (-1)^i (a_m - a_{m-1}) s_{m 1^i}.$

As the Schur polynomials are orthonormal on the unitary group, the claim follows. $\Box$

The CUE hypothesis would then imply the corresponding mollified moment conjecture

$\displaystyle {\bf E}_{Q} |M(U) P(1)|^2 = |a_0|^2 + N \sum_{m=1}^d |a_m - a_{m-1}|^2 + o(1). \ \ \ \ \ (14)$

(See this paper of Conrey, and this paper of Radziwill, for some discussion of the analogous conjecture for the zeta function, which is essentially due to Farmer.)

From Proposition 3 one sees that this conjecture holds in the range ${d \leq \frac{1}{2} N}$. It is likely that the function field analogue of the calculations of Conrey (based ultimately on deep exponential sum estimates of Deshouillers and Iwaniec) can extend this range to ${d < \theta N}$ for any ${\theta < \frac{4}{7}}$, if ${N}$ is sufficiently large depending on ${\theta}$; these bounds thus go beyond what is available from Proposition 3. On the other hand, as discussed in Remark 7 of the previous post, ACUE would also predict (14) for ${d}$ as large as ${N-2}$, so the available mollified moment estimates are not strong enough to rule out ACUE. It would be interesting to see if there is some other estimate in the function field setting that can be used to exclude the ACUE hypothesis (possibly one that exploits the fact that GRH is available in the function field case?).