Hypergeometric rational approximations to ζ(4)

Abstract We give a new hypergeometric construction of rational approximations to ζ(4), which absorbs the earlier one from 2003 based on Bailey's 9F8 hypergeometric integrals. With the novel ingredients we are able to gain better control of the arithmetic and produce a record irrationality measure for ζ(4).


Introduction
Apéry's proof [1, 6, 18] of the irrationality of ζ (3) in the 1970s sparked research in arithmetic on the values of Riemann's zeta function ζ(s) at integers s ≥ 2. Some particular representatives of this development include [4,8,9,13], and the story culminated in a remarkable arithmetic method [14,15] of Rhin and Viola to produce sharp irrationality measures for ζ (2) and ζ(3) using groups of transformations of rational approximations to the quantities. In spite of hopes to (promptly) extend Apéry's success to ζ (5) and other zeta values, the next achievement in this direction [3,16] materialized only in the 2000s in the work of Ball and Rivoal. The latter result helped to unify different-looking approaches for arithmetic investigations of zeta values ζ(s) and related constants under a 'hypergeometric' umbrella, with some particular highlights given in [19,20] by one of these authors. The hypergeometric machinery has proven to be useful in further arithmetic applications; see, for example, [7,11,12,22] for more recent achievements.
The quantity ζ (4), though known to be irrational and even transcendental, remains a natural target for testing the hypergeometry. Apéry-type approximations to the number were discovered and rediscovered on several occasions [5,17,19], but they were not good enough to draw conclusions about its irrationality. In [19], a general construction of rational approximations to ζ(4) is proposed, which makes use of very-well-poised hypergeometric integrals and a group of their transformations; this leads to an estimate for the irrationality exponent of the number in question provided that a certain 'denominator conjecture' for the rational approximations is valid. The conjecture appears to be difficult enough, with its only special case established in [10] but insufficient for arithmetic applications. This case is usually dubbed 'most symmetric', because the group of transformations acts trivially on the corresponding approximations.
The principal goal of this work is to recast the rational approximations to ζ(4) from [19] in a different (but still hypergeometric) form and obtain, by these means, better control of the arithmetic of their coefficients. In this way, we are able to produce the estimate μ(ζ(4)) 12.51085940 . . .
for the irrationality exponent of the zeta value, which is better than the conjectural one given in [19]. This is not surprising, as we do not attempt to prove the denominator conjecture from [19] but instead investigate the arithmetic of approximations from a different hypergeometric family.
The plan of our exposition below is as follows. In § 2 we give a Barnes-type double integral for rational approximations to ζ(4) and then, in § 3, work out the particular 'most symmetric' case of this integral, which clearly illustrates arithmetic features of the new representation of the approximations. We recall general settings from [19] in § 4 and embed the approximations into a 12-parametric family of hypergeometric-type sums that are further discussed in greater detail in § 5. Furthermore, § 6 reviews (and recovers) the permutation group related to the linear forms in 1 and ζ(4) from a special subfamily of the approximations constructed. Finally, we investigate arithmetic aspects of the general rational approximations in § 7 and produce a calculation that leads to the new bound for μ(ζ(4)) in § 8.
In the text below, we intentionally avoid producing claims (in the form of propositions and lemmas), to give our exposition the nature of storytelling rather than traditional mathematical writing.

Integral representations
For k ≥ 2 even, fix a generic set of complex parameters and define as in [21] the very-well-poised hypergeometric integrals dt.
By Bailey's integral analogue of Dougall's theorem [2, § 6.6], Substituting this into the iteration the latter can be given as (1)
We first deal with the internal integral in (2). The rational integrand is decomposed into the sum of partial fractions: Then by the residue sum theorem. The choices ν 0 = 0 and ν 0 = −n lead to the equality since A k (t) are polynomials, the two representations imply that the only poles of H n (t) are located at the integers Furthermore, the function has poles only at t = −(n + 1), −(n + 2), . . . , −(2n + 1) and vanishes at t = −1, −2, . . . , −n. Moreover, H n (t) is in fact a rational function of degree at most −2 (so that it has the zero residue at infinity); indeed, it is the sum of rational functions each of degree at most −2 (in t). This means that we have a partial-fraction decomposition With the help of the following consequence of formula (3), we find that and similarly Note that for all j, k ∈ Z by the standard arithmetic properties of integer-valued polynomials [22,Lemma 4], where d n denotes the least common multiple of 1, 2, . . . , n. Furthermore, each term of the sums for B j and C j has a factor of the form and these quantities are all divisible by the greatest common divisor Φ n of numbers (there are only finitely many non-zero products on the list). Thus, This implies that In § 7 we reveal details of the computation of Φ n (and its asymptotics as n → ∞); we show that Φ n is divisible by the product over primes This corresponds to the 'denominator conjecture' from [19]; for the most symmetric case in this section, it was established earlier in [10] using different hypergeometric techniques.

Old approximations to ζ(4)
We now concentrate on a specific setting of § 2, where k = 6 and the parameters are positive integers satisfying the conditions Define the rational function Then (4) with any t 0 ∈ Z, 1 − min is essentially the very-well-poised hypergeometric integral given in [19]. Notice, however, that the arithmetic normalization factor γ(h) differs slightly from the one used in [19].
Rearranging the order of parameters in (1) we obtain The double integral we arrive at belongs to a more general (12-parametric) family, which we discuss in the next section.
Note that simultaneous shifts of a 0 , a 1 , a 3 , a 5 and b 0 , b 1 , b 3 , b 5 by the same integer do not affect G (a, b); the same is true for simultaneous shifts of a 0 , a 2 , a 4 , a 6 and b 0 , b 2 , b 4 , b 6 . (In particular, the shifts by given 1 − b 1 and 1 − b 2 , respectively, allow us to assume that b 1 = b 2 = 1.) The latter two symmetries potentially leave 12 out of 14 parameters (7) independent. Furthermore, we choose to be a reordering of the parameters (7) (so that (9) and (7) coincide as multisets) such that Similar to the most symmetric case in § 3, we may choose the integration paths in (6) to be the vertical lines {c 1 + iy : y ∈ R} for s and {c 2 + iy : y ∈ R} for t, with and we take c 1 = 1/3 − a * 1 and c 2 = 1/3 − a * 2 . Also, the rational function in s and t at the integrand in (6) has degree at most −2 both in s and in t, and the functions 1 sin πs , cos πs (sin πs) 2 and 1 sin πt are bounded in their respective integration domains. By sin π(s + t) = sin πs cos πt + cos πs sin πt, G(a, b) is split into two absolutely convergent integrals, and, after interchanging the order of integrations in s and in t in the second integral, we obtain ds dt + a similar integral with a j , b j changed to a 7−j , b 7−j for j = 1, . . . , 6. (11) As already seen in the most symmetric case, the integral H(a 0 , a 1 , a 3 , a 5 is a rational function in t, and we may even vary c 1 in the interval −a * 3 < c 1 < 1 − b * 3 , because a power of sin πs is dropped in the denominator of (12) with respect to the integral (6). In executing this, we do not have to take care of possible poles coming from (t + s + a 0 ) b0−a0 , because it never vanishes if t is chosen in an appropriate region of the complex plane, and two rational functions that coincide in such a region must coincide everywhere.
Explicitly, we have Then where ν 0 is any integer in the interval 1 − a * Since all A k (t) are polynomials, the poles of function (14) are only possible at For a similar reason, with ν 0 in the larger interval 1 − a * 5 ν 0 1 − b * 1 , the function Since it follows that the set of double poles of H(t) coincides with the set of simple poles of I(t) and therefore is also contained at integers in [a *

however, H(t) may still possess simple poles at integers in [a
Arguing as in § 3, we arrive at the partial-fraction decomposition because the rational function H(t) has degree at most −2 by (8). Noticing that the has at least simple zeroes at t = 1 − a * 6 , . . . , −b * 2 and at least double zeroes at t = 1 − a * 4 , 2 − a * 4 , . . . , −b * 4 , and taking into account condition (10), we find that H(t) does not have poles in the half-plane Re t > c 2 ; hence, the expansion (15) 'shortens' to In fact, the second sum is over the interval max{a * − 1 and may be even empty if the interval is empty. With the explicit expressions (13) and (14) (used, for example, with ν 0 = 1 − a * 3 ) in mind, we conclude that the coefficients [20,Lemmas 17,18] where j C j = 0 is implemented. Proceeding in the same way for the second double integral in (11), we conclude that G (a, b) = B(a, b)ζ(4) − C(a, b) with a 1 − b 1 , . . . , a 6 − b 6 }, and ord p B, 4 + ord p C ≥ min for primes p > √ b 0 − a 0 − 2. Finally, we remark that condition (10) is conventional (and happens to hold in our applications, even in the form of equality b * 3 + b * 4 = a 0 − 1) but can potentially be dropped without significant arithmetic losses. For example, if b * 1 + b * 2 > a 0 − 1 then the partialfraction decomposition (15) translates into so that there are poles of H(t) to the right of the contour Re t = c 2 . The corresponding residues of the integrand are where j is an integer in the interval 1 + a 0 − b * 3 j b * 4 − 1 and we use the expansion π sin πt 3 cos πt = 1 (t + j) 3 − 6ζ(4)(t + j) + O (t + j) 3 as t → −j.
Proceeding as above we deduce that which again can be seen to be a linear form in Zζ(4) + Q.

The group structure for ζ(4)
Following [19], to any set of parameters h from § 4 we assign the 27-element multiset of non-negative integers and set H(e) = F (h) for the quantity defined in that section. By construction, is invariant under any permutation of the parameters h 1 , h 2 , . . . , h 6 (which we can view as the 'h-trivial' action). Clearly, any such permutation induces the corresponding permutation of the parameter set (18). On the other hand, it can be seen from (6) that the quantity does not change when the parameters in either collection a 1 , a 3 , a 5 or a 2 , a 4 , a 6 permute; we can regard such permutations as 'a-trivial'. (The same effect is produced by 'b-trivial' permutations, when we change the order in b 1 , b 3 , b 5 or b 2 , b 4 , b 6 .) We can also add to the list the 'trivial' involution i : a j ↔ a 7−j , b j ↔ b 7−j for j = 1, . . . , 6, which reflects the symmetry s ↔ t of the double integral (6). In addition, we recall that G (a, b) is left unchanged by the simultaneous shifts of a 0 , a 1 , a 3 , a 5 and b 0 , b 1 , b 3 , b 5 (or of a 0 , a 2 , a 4 , a 6 and b 0 , b 2 , b 4 , b 6 , respectively) by the same integer. We regard the action of all these transformations (permutations, shifts and involution) and their compositions as the '(a, b)-trivial' action. By setting we have F (h) = G(a, b). If we request that the condition holds, then the shift of h 0 , h 1 , h 3 , h 5 by 1 induces the composition of the shift of a 0 , a 1 , a 3 , a 5 and b 0 , b 1 , b 3  on the parameter set (18), is an (a, b)-trivial transformation. As a consequence, the quantity G(a, b) = e 01 !e 02 !e 05 !e 06 !e 15 !e 26 !H(e) does not change by the action of the permutation b. We remark that (20) is a very natural condition for the application of the (a, b)-trivial action to F (h). Indeed, by (19) under the permutation b. The permutation group of the multiset (18), which is generated by all h-trivial permutations and the permutation b, coincides with the group G (of order 51840) considered in [19]. (Note that the group contains the above involution i as well.) By these means, we also recover the invariance of the quantity  G(a, b)-representation, we are interested in collecting a set of representatives which are distinct modulo (a, b)-trivial transformations. For a generic set of integral parameters h subject to (20), such a set of representatives contains 120 different elements. Indeed, by (19) and (20), the subgroup of all the (a, b)-trivial permutations in G contains 3! 3 · 2! = 432 elements and is generated by: • the aand h-trivial permutations (h 1 h 3 ) and (h 3 h 5 ); • the aand h-trivial permutations (h 2 h 4 ) and (h 4 h 6 ); • the b-trivial permutation (b 1 b 3 )(b 4 b 6 ) (that is, by b 135 ); and • the involution i (that is, by (h 1 h 6 )(h 2 h 5 )(h 3 h 4 )).