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A Recursion Formula for Moments of Derivatives of Random Matrix Polynomials

We give asymptotic formulae for random matrix averages of derivatives of characteristic polynomials over the groups USp(2N), SO(2N) and O^-(2N). These averages are used to predict the asymptotic formulae for moments of derivatives of L-functions which arise in number theory. Each formula gives the leading constant of the asymptotic in terms of determinants of hypergeometric functions. We find a differential recurrence relation between these determinants which allows the rapid computation of the (k+1)-st constant in terms of the k-th and (k-1)-st. This recurrence is reminiscent of a Toda lattice equation arising in the theory of \tau-functions associated with Painlev\'e differential equations.


Introduction
For over 50 years, mathematicians and physicists have used random matrix theory to study a wide-ranging, growing list of probabilistic phenomena. Particularly surprising are its applications in number theory, where random matrices model the distribution of nontrivial zeros of the Riemann zeta function. Random matrix theory now provides far-reaching and widely believed conjectures for many questions in the analytic theory of L-functions.
Katz and Sarnak [KaSa] give evidence that every family of L-functions falls into one of four symmetry types: unitary U(N), unitary symplectic USp(2N), even orthogonal SO(2N) and odd orthogonal O − (2N). These symmetry types govern the distribution of zeroes and special values in families. Using random matrix models, Keating and Snaith [KeSn] and Conrey,Farmer,Keating,Rubinstein and Snaith [CFKRS03] have produced deep conjectures for estimating the integral moments of central values in families of L-functions.
The derivatives of L-functions are also of great interest, and are the subject of this paper. A motivational example is Speiser's theorem, which asserts that the Riemann hypothesis is equivalent to the nonexistence of nonreal zeros of the derivative of the Riemann zeta function to the left of the critical line, see e.g. [Sou]. Moreover, the derivatives of L-functions control the order of vanishing at the central point, which encodes important arithmetic and geometric information. For example, according to the Birch and Swinnerton-Dyer Conjecture, the order of vanishing of the L-function of an elliptic curve over the rationals coincides with the arithmetic rank of the curve.
An L-function is modeled by the characteristic polynomial Λ A of a random matrix A. Here we compute where G denotes USp, SO, or O − , and dA is the Haar measure on G. As N → ∞, this models the kth moment of L (m) (1/2) in a family of symmetry type G.

1
One can find the moments of Λ (m) A (1) by differentiating the corresponding shifted moment formulae, which are computed in [CFKRS03]. Conrey, Rubinstein and Snaith in [CRS] develop a faster method to compute the relevant averages in the unitary case: This leading constant b k is the same "geometric constant" appearing in conjectures for the asymptotic estimate of the k-th moment of ζ ′ (s) in t aspect. Conrey, Rubinstein and Snaith describe b k in terms of a k × k determinant of I-Bessel functions, and are able to compute b k numerically for k ≤ 15. Forrester and Witte in [FW06,FW02] find a surprising expression for these determinants of I-Bessel functions in terms of solutions to Painlevé III ′ differential equations. Explicitly, the formula in [CRS] is as follows: where I ℓ (x) is the modified Bessel function of the first kind. One then defines Forrester and Witte [FW06,FW02] find that this τ k (t) (denoted τ [k](t) in [FW02,section 4]) is in fact the Okamoto τ function associated with the Painlevé III ′ differential equation: This nonlinear second order differential equation has a solution with certain boundary data (see [FW02]) given in terms of τ k (t) by the formula Specifying boundary conditions, one can quickly compute σ III,k (t) from the differential equation and recover τ k (t) via the equation This expression allows a much faster computation of the constants b k . The goal of the present paper is to extend these results to the other symmetry types relevant to L-functions: USp(2N), SO(2N) and O − (2N).
Employing similar techniques to [CRS], we obtain analogous results, where the role of the I-Bessel functions above is here played by hypergeometric functions, for u ∈ C and m ∈ Z. For negative m, interpret the above expression as the limit. The role of the τ -function is played by for k ≥ 0, ℓ ∈ Z and u ∈ C, where, here and in the following, the indices i and j of the matrix in the determinant range from 1 to k. In the context of Theorems 1, 2 and 3 below, the ℓ appearing here takes values 0, −1, 0 respectively. We now state theorems for the symplectic, special orthogonal, and negative orthogonal cases: We are naturally led to consider the second and third derivatives of characteristic polynomials in the above theorems instead of the first derivative due to root number considerations. Indeed, if A is a unitary matrix, the characteristic polynomial satisfies the functional equation and one has a simple expression for Λ ′ A (1) in terms of Λ A (1), which can be used to compute the moments of the derivative via partial integration. Thus the moments of Λ ′′ A (1) give the next novel information. When A ∈ O − (2N) we have det A = −1, and thus Λ A (1) = 0. In this case Λ ′ A (1) plays the role which Λ A (1) plays in the other families, Λ ′′ A (1) has a simple expression in terms of Λ ′ A (1), and therefore one considers moments of Λ ′′′ A (1). The same reasoning carries over to the families of L-functions having each of the aforementioned symmetry types.
The method of proof of Theorems 1, 2 and 3 can be generalized to higher-order derivatives easily. It suffices to expand the binomial in Lemma 4, and otherwise proceed as in the given proofs of Theorems 1, 2 and 3.
We can use Theorems 1, 2 and 3 to give conjectures for moments of derivatives of Lfunctions at s = 1/2 with the above symmetry types. For example, the quadratic Dirichlet L-functions ordered by conductor form a symplectic family and thus we make the following conjecture: Conjecture 1. Let D(X) := {|d| < X, d fundamental discriminant}, and L(s, χ d ) denote the quadratic Dirichlet L-function of fundamental discriminant d. The average value of the second derivative of quadratic Dirichlet L-functions at the central point is Here a k is a well understood arithmetic constant depending on the functional equations in the family: . This is the same arithmetic constant appearing in moment conjectures for L(1/2, χ d ) without the derivative. It is not predicted by random matrix calculations.
Our Theorems 1, 2 and 3 along with the results of [CRS] allow similar conjectures to be made for any family of L-functions.
Under closer examination, the determinants T k,ℓ (u) defined by (1) and appearing in Theorems 1, 2 and 3 exhibit a surprisingly rich structure. Our Theorem 4 is a differential recurrence relation which allows much faster computation of the constants b k (G(N), m): This recurrence relation closely resembles a Toda lattice equation for the Okamoto τfunction associated with a Painlevé differential equation, see [Oka,Theorem 2]. Such Toda lattice equations are at the heart of the τ -function theory of Painlevé equations, and are used by Forrester and Witte [FW06,FW02] to connect determinants the of I-Bessel functions found by [CRS] to the Painlevé III ′ equation. It would be very interesting to determine whether or not there exists a differential equation arising from our formula (5) which plays the role for symplectic and orthogonal types that Painlevé III ′ plays for unitary symmetry.
Ultimately, one hopes to obtain formulae for the complex moments of characteristic polynomials in the various symmetry types. In the case of the undifferentiated moment conjectures, it has been found that the geometric constants g k can be expressed in a simple form in terms of Barnes G-functions, which are well-defined for complex values of k, see [CF]. A project for the future would be to see if there exists a similar expression for the geometric coefficients b k studied in this paper for moments of derivatives of L-functions.
Acknowledgements: We would like to thank the organizers and staff of the American Mathematical Society's Mathematics Research Community program on Arithmetic Statistics in Snowbird, Utah, who made our work possible. We thank the staff of the American Institute of Mathematics for their hospitality. Particular thanks to Brian Conrey for his guidance throughout this project. Λ(z 1 ) · · · Λ(z k ) dA.
and the lemma follows.
Our proofs will also rely upon Vandermonde determinants of differential operators: We give two lemmas on computing with these-Lemma 5 is a direct consequence of the definition, but we prove Lemma 6 in detail.
Lemma 6. Let f (x, y) be k − 1 times differentiable in x and y. Then where the sum runs over the permutations µ ∈ S k . Applying Lemma 5 again, we find Now, we may rearrange the rows of the matrix doing so cancels out the sign(µ) attached to the determinant, and we reach the desired formula.
Our final lemma is a recursion for determinants discovered by Lewis Carroll. It will be used in the proof of Theorem 4.
Lemma 7. Let A be an k × k matrix, and let A a 1 ,...,ar b 1 ,...,bs denote the matrix A with rows a 1 , . . . , a r and the columns b 1 , . . . , b s removed. Then

8
Thus we may proceed towards Theorems 1, 2, and 3 by differentiating the formulae found in Corollaries 1, 2, and 3, respectively. The asymptotics remain valid after differentiating because they are uniform in α i .
Proof of Theorem 1. From Corollary 1 and the above argument, we know that the kth moment of Λ ′′ is asymptotically We apply Lemma 4 with m = 2, finding This allows us to writẽ .
Proof of Theorem 2. We can proceed in the same way as in the proof of Theorem 1, starting from Corollary 2.
Proof of Theorem 3. This time we consider the third derivative. From (6) we have