Hankel Determinants of Zeta Values

We study the asymptotics of Hankel determinants constructed using the values $\zeta(an+b)$ of the Riemann zeta function at positive integers in an arithmetic progression. Our principal result is a Diophantine application of the asymptotics.


Introduction
In the recent work [4], H. Monien  where ζ(s) denotes the Riemann zeta function. He also studied more general determinants constructed using values of Dirichlet series. One focus of that work was the asymptotic behaviour of H (0) n and H (1) n as n → ∞, and a heuristic justification for the simplified asymptotic formula log H (r) n = −n 2 log n + O n 2 as n → ∞ for r ≥ 0.
In [5] Monien developed these ideas further and rigorously justified the above asymptotics in the case r = 0, by explicitly constructing a family of orthogonal polynomials related to the corresponding Riemann-Hilbert problem. However, his approach does not readily generalize to prove the expected asymptotics for determinants built on the zeta values ζ(an + b) along an arithmetic progression, which are more interesting from an arithmetical point of view. To be precise, for positive integers a and b, we expect that log det 1≤i,j≤n ζ(a(i + j) + b) = −an 2 log n + O n 2 as n → ∞.
This type of result is of interest, for example, when a is even and b is odd, so that the obviously irrational (and transcendental) zeta values at positive even integers are excluded.
In this brief note we demonstrate how elementary means can be used to prove the weaker asymptotic inequality log det 1≤i,j≤n ζ(a(i + j) + b) ≤ − a 2 n 2 log n + O n 2 as n → ∞, which leads us to the following arithmetic application.
Theorem 1. For any pair of positive integers a and b, either there are infinitely many n ∈ N for which ζ(an + b) is irrational, or the sequence {q n } ∞ n=1 of common denominators of the rational elements of the set {ζ(a + b), ζ(2a + b), . . . , ζ(an + b)} grows super-exponentially, i.e., q The current knowledge about the arithmetic of odd zeta values -the numbers ζ(n) for n ≥ 2 odd -can be summarised as follows. In 1978 R. Apéry proved [1] the irrationality of ζ(3), and in 2000 K. Ball and T. Rivoal showed [2] that there are infinitely many irrational numbers among the odd zeta values. At the same time, it is widely believed that all of the numbers ζ(n), for n ≥ 2, are irrational and transcendental. Our result in Theorem 1 is very far from proving this. The goal of this paper is, rather, to demonstrate how very simple analytic arguments can be used to derive some information about the Diophantine approximation properties of these numbers.

Asymptotic upper bounds
Suppose that {a n } ∞ n=1 is a sequence of complex numbers which satisfies a n ≪ n 1−δ for some δ > 0. Then the Dirichlet series f (s) = ∞ n=1 a n n s converges in the region Re(s) > σ 0 for some σ 0 > 2 − δ. For each n ∈ Z >0 , let 1≤i,j≤n f (i + j) . Lemma 1. As n → ∞, the following estimate is valid: Proof . Using the linearity of determinant with respect to each row and the formula for the Vandermonde determinant we have that Now by considering one of the n! possible orderings of the integers k 1 , . . . , k n , and using our assumption on the numbers a n we obtain that where the estimate is invoked. Finally, taking logarithms gives the desired result.
As a slight generalization of the above argument, let a and b be positive integers and consider the Hankel determinants By the same steps as before, we obtain that and taking logarithms gives us the following result.

Lemma 2.
As n → ∞, the following estimate is valid:

Concluding remarks
The same techniques used to prove Theorem 1 can also be applied to a much broader class of Dirichlet series. For example, similar results can be obtained for more Dirichlet L-functions and even L-functions attached to modular forms, by extending the argument in the proof of Lemma 3. In fact, one does not even need to restrict to values of these functions at integers. In a different direction, one can deal with the values at subsequences which tend to infinity faster than arithmetic progressions (for example, the sequence ζ(2n 2 + 1), where n = 1, 2, . . . ). The estimates from Section 2 would then become sharper, and the corresponding growth of the common denominators, provided infinitely many of the L-values are rational, can be then shown to be faster. Of course, as mentioned in the introduction, we expect that the nonzero zeta and L-function values at positive integers are always irrational (and even transcendental). Our principal result here is only to serve as an illustration of a much deeper relationship between Hankel determinants and arithmetic.