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This is another sequel to a recent post in which I showed the Riemann zeta function can be locally approximated by a polynomial, in the sense that for randomly chosen one has an approximation

where grows slowly with , and is a polynomial of degree . 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 -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 -functions is certainly well studied). In this model it is possible to set fixed and let 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 of some order and a natural number , and set

We will primarily think of as being large and as being either fixed or growing very slowly with , though it is possible to also consider other asymptotic regimes (such as holding fixed and letting go to infinity). Let be the ring of polynomials of one variable with coefficients in , and let be the multiplicative semigroup of monic polynomials in ; one should view and as the function field analogue of the integers and natural numbers respectively. We use the valuation for polynomials (with ); this is the analogue of the usual absolute value on the integers. We select an irreducible polynomial of size (i.e., has degree ). The multiplicative group can be shown to be cyclic of order . A Dirichlet character of modulus is a completely multiplicative function of modulus , that is periodic of period and vanishes on those not coprime to . From Fourier analysis we see that there are exactly Dirichlet characters of modulus . A Dirichlet character is said to be *odd* if it is not identically one on the group of non-zero constants; there are only non-odd characters (including the principal character), so in the limit 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 be an odd Dirichlet character of modulus . The Dirichlet -function is then defined (for of sufficiently large real part, at least) as

Note that for , the set is invariant under shifts whenever ; since this covers a full set of residue classes of , and the odd character has mean zero on this set of residue classes, we conclude that the sum vanishes for . In particular, the -function is entire, and for any real number and complex number , we can write the -function as a polynomial

where and the coefficients are given by the formula

Note that can easily be normalised to zero by the relation

In particular, the dependence on is periodic with period (so by abuse of notation one could also take to be an element of ).

Fourier inversion yields a functional equation for the polynomial :

Proposition 1 (Functional equation)Let be an odd Dirichlet character of modulus , and . There exists a phase (depending on ) such thatfor all , or equivalently that

where .

*Proof:* We can normalise . Let be the finite field . We can write

where denotes the subgroup of consisting of (residue classes of) polynomials of degree less than . Let be a non-trivial character of whose kernel lies in the space (this is easily achieved by pulling back a non-trivial character from the quotient ). We can use the Fourier inversion formula to write

where

From change of variables we see that is a scalar multiple of ; from Plancherel we conclude that

for some phase . We conclude that

The inner sum equals if , and vanishes otherwise, thus

For in , and the contribution of the sum vanishes as is odd. Thus we may restrict to , so that

By the multiplicativity of , this factorises as

From the one-dimensional version of (3) (and the fact that is odd) we have

for some phase . The claim follows.

As one corollary of the functional equation, is a phase rotation of and thus is non-zero, so has degree exactly . The functional equation is then equivalent to the zeroes of being symmetric across the unit circle. In fact we have the stronger

Theorem 2 (Riemann hypothesis for Dirichlet -functions over function fields)Let be an odd Dirichlet character of modulus , and . Then all the zeroes of 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 ), we may write

for some unitary matrix . It is possible to interpret 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 arising from very small primes is now absent (in the function field setting there are no primes of size between and ).

We now let vary uniformly at random over all odd characters of modulus , and uniformly over , independently of ; we also make the distribution of the random variable conjugation invariant in . We use to denote the expectation with respect to this randomness. One can then ask what the limiting distribution of is in various regimes; we will focus in this post on the regime where is fixed and is being sent to infinity. In the spirit of the Sato-Tate conjecture, one should expect to converge in distribution to the circular unitary ensemble (CUE), that is to say Haar probability measure on . 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 be the eigenvalues of (ordered arbitrarily), then we now have

and hence the are essentially elementary symmetric polynomials of the eigenvalues:

One can take log derivatives to conclude

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

where has sufficiently large real part, , and the von Mangoldt function is defined as when is the power of an irreducible and otherwise. We conclude the “explicit formula”

Similarly on inverting we have

Since we also have

for sufficiently large real part, where the Möbius function is equal to when is the product of distinct irreducibles, and otherwise, we conclude that the Möbius coefficients

are just the complete homogeneous symmetric polynomials of the eigenvalues:

One can then derive various algebraic relationships between the coefficients from various identities involving symmetric polynomials, but we will not do so here.

What do we know about the distribution of ? By construction, it is conjugation-invariant; from (2) it is also invariant with respect to the rotations for any phase . We also have the function field analogue of the Rudnick-Sarnak asymptotics:

Proposition 3 (Rudnick-Sarnak asymptotics)Let be nonnegative integers. Ifis equal to in the limit (holding fixed) unless for all , in which case it is equal to

Comparing this with Proposition 1 from this previous post, we thus see that all the low moments of are consistent with the CUE hypothesis (and also with the ACUE hypothesis, again by the previous post). The case of this proposition was essentially established by Andrade, Miller, Pratt, and Trinh.

*Proof:* We may assume the homogeneity relationship

since otherwise the claim follows from the invariance under phase rotation . By (6), the expression (9) is equal to

where

and consists of copies of for each , and similarly consists of copies of for each .

The polynomials and are monic of degree , which by hypothesis is less than the degree of , and thus they can only be scalar multiples of each other in if they are identical (in ). As such, we see that the average

vanishes unless , in which case this average is equal to . Thus the expression (9) simplifies to

There are at most choices for the product , and each one contributes to the above sum. All but of these choices are square-free, so by accepting an error of , we may restrict attention to square-free . This forces to all be irreducible (as opposed to powers of irreducibles); as is a unique factorisation domain, this forces and to be a permutation of . By the size restrictions, this then forces for all (if the above expression is to be anything other than ), and each is associated to possible choices of . Writing and then reinstating the non-squarefree possibilities for , we can thus write the above expression as

Using the prime number theorem , we obtain the claim.

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

Corollary 4 (CUE statistics at low frequencies)Let be the eigenvalues of , permuted uniformly at random. Let be a linear combination of monomials where are integers with either or . Then

The analogue of the GUE hypothesis in this setting would be the CUE hypothesis, which asserts that the threshold here can be replaced by an arbitrarily large quantity. As far as I know this is not known even for (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 , though not to .

*Proof:* By permutation symmetry we can take to be symmetric, and by linearity we may then take to be the symmetrisation of a single monomial . If then both expectations vanish due to the phase rotation symmetry, so we may assume that and . We can write this symmetric polynomial as a constant multiple of plus other monomials with a smaller value of . Since , the claim now follows by induction from Proposition 3 and Proposition 1 from the previous post.

Thus, for instance, for , the moment

is equal to

because all the monomials in are of the required form when . The latter expectation can be computed exactly (for any natural number ) using a formula

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

and more generally

when , where are the integers

and more generally

(OEIS A039622). Thus we have

for if and is sufficiently slowly growing depending on . The CUE hypothesis would imply that that this formula also holds for higher . (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 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

where

is some finite linear combination of monomials independent of . We can expand the above sum as

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

This is the analogue of Montgomery’s pair correlation conjecture. Proposition 3 implies that this claim is true whenever is supported on . If instead we assume the ACUE hypothesis (or the weaker Alternative Hypothesis that the phase gaps are non-zero multiples of ), one should instead have

for arbitrary ; this is the function field analogue of a recent result of Baluyot. In any event, since is non-negative, we unconditionally have the lower bound

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

Then (12) applies unconditionally and we conclude that

The right-hand side evaluates to . On the other hand, is non-negative, and equal to when . Thus

The sum is at least , and is at least if is not a simple eigenvalue. Thus

and thus the expected number of simple eigenvalues is at least ; 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 , the same arguments allow one to show that the expected proportion of simple eigenvalues is .

Suppose that the phase gaps in are all greater than almost surely. Let is non-negative and non-positive for outside of the arc . Then from (13) one has

so by taking contrapositives one can force the existence of a gap less than asymptotically if one can find with non-negative, non-positive for outside of the arc , and for which one has the inequality

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

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

where

for some coefficients . We can compute this moment in the CUE case:

Proposition 5We have

*Proof:* From (5) one has

hence

where we suppress the dependence on the eigenvalues . Now observe the Pieri formula

where are the hook Schur polynomials

and we adopt the convention that vanishes for , or when and . Then also vanishes for . We conclude that

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

The CUE hypothesis would then imply the corresponding mollified moment conjecture

(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 . 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 for any , if is sufficiently large depending on ; 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 as large as , 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?).

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