Hypergeometric rational approximations to $\zeta(4)$

TL;DR

This paper introduces a new hypergeometric approach to approximate ζ(4), improving the control over arithmetic properties and establishing a record irrationality measure for this value.

Contribution

It presents a novel hypergeometric construction that enhances previous methods and achieves a better irrationality measure for ζ(4).

Findings

New hypergeometric construction for ζ(4)

Improved irrationality measure for ζ(4)

Enhanced control of arithmetic properties

Abstract

We give a new hypergeometric construction of rational approximations to $\zeta(4)$, which absorbs the earlier one from 2003 based on Bailey's ${}_9F_8$ hypergeometric integrals. With the novel ingredients we are able to get a better control of arithmetic and produce a record irrationality measure for $\zeta(4)$.

Equations

$$\mu (\zeta (4))⩽12.51085940\dots$$

$$\mathbf{h}=(h_0,h_{-1};h_1,h_2,\dots ,h_k)$$

$$\max \{0,\mathrm{Re}(h_0-h_{-1})\}<\mathrm{Re}{h}_j<\frac{1}{2}\mathrm{Re}{h}_0\text{for}j=1,\dots ,k,$$

$$F_k^{\prime }(\mathbf{h})$$

$$=\frac{1}{2\pi i}{\int }_{-i\infty }^{i\infty }(h_0+2t)\frac{{\prod }_{j=-1}^k\Gamma (h_j+t)\cdot \Gamma (h_{-1}-h_0-t)\Gamma (-t)}{{\prod }_{j=1}^k\Gamma (1+h_0-h_j+t)}\mathrm{d}t.$$

$$F_2^{\prime }(h_0,h_{-1};h_1,h_2)=\frac{\Gamma (h_{-1})\Gamma (h_1)\Gamma (h_2)\Gamma (h_1+h_{-1}-h_0)\Gamma (h_2+h_{-1}-h_0)}{\Gamma (1+h_0-h_1-h_2)\Gamma (h_1+h_2+h_{-1}-h_0)}.$$

$$F_{k+2}^{\prime }(h_0,h_{-1};h_1,\dots ,h_{k-1},h_k,h_{k+1},h_{k+2})$$

$$=\frac{1}{\Gamma (1+h_0-h_k-h_{k+1})\Gamma (1+h_0-h_k-h_{k+2})\Gamma (1+h_0-h_{k+1}-h_{k+2})}$$

$$\times \frac{1}{2\pi i}{\int }_{-i\infty }^{i\infty }\Gamma (h_k+s)\Gamma (h_{k+1}+s)\Gamma (h_{k+2}+s)$$

$$\times \Gamma (1+h_0-h_k-h_{k+1}-h_{k+2}-s)\cdot F_k^{\prime }(h_0,h_{-1};-s,h_1,\dots ,h_{k-1})\mathrm{d}s$$

$$F_4^{\prime }(h_0,h_{-1};h_1,h_2,h_3,h_4)$$

$$=\frac{\Gamma (h_{-1})\Gamma (h_1)\Gamma (h_1+h_{-1}-h_0)}{\Gamma (1+h_0-h_2-h_3)\Gamma (1+h_0-h_2-h_4)\Gamma (1+h_0-h_3-h_4)}$$

$$\times \frac{1}{2\pi i}{\int }_{-i\infty }^{i\infty }\Gamma (h_2+s)\Gamma (h_3+s)\Gamma (h_4+s)\Gamma (1+h_0-h_2-h_3-h_4-s)$$

$$\times \frac{\Gamma (h_{-1}-h_0-s)\Gamma (-s)}{\Gamma (1+h_0-h_1+s)\Gamma (h_1+h_{-1}-h_0-s)}\mathrm{d}s$$

$$F_6^{\prime }(h_0,h_{-1};h_1,h_2,h_3,h_4,h_5,h_6)$$

$$=\frac{1}{\Gamma (1+h_0-h_4-h_5)\Gamma (1+h_0-h_4-h_6)\Gamma (1+h_0-h_5-h_6)}$$

$$\times \frac{1}{2\pi i}{\int }_{-i\infty }^{i\infty }\Gamma (h_4+t)\Gamma (h_5+t)\Gamma (h_6+t)$$

$$\times \Gamma (1+h_0-h_4-h_5-h_6-t)\cdot F_4^{\prime }(h_0,h_{-1};-t,h_1,h_2,h_3)\mathrm{d}t$$

$$=\frac{1}{\Gamma (1+h_0-h_4-h_5)\Gamma (1+h_0-h_4-h_6)\Gamma (1+h_0-h_5-h_6)}$$

$$\times \frac{\Gamma (h_{-1})}{\Gamma (1+h_0-h_1-h_2)\Gamma (1+h_0-h_1-h_3)\Gamma (1+h_0-h_2-h_3)}$$

$$\times \frac{1}{2\pi i}{\int }_{-i\infty }^{i\infty }\Gamma (h_4+t)\Gamma (h_5+t)\Gamma (h_6+t)\Gamma (1+h_0-h_4-h_5-h_6-t)$$

$$\times \frac{1}{2\pi i}{\int }_{-i\infty }^{i\infty }\Gamma (h_1+s)\Gamma (h_2+s)\Gamma (h_3+s)\Gamma (1+h_0-h_1-h_2-h_3-s)$$

$$\times \frac{\Gamma (h_{-1}-h_0-t)\Gamma (-t)\Gamma (h_{-1}-h_0-s)\Gamma (-s)}{\Gamma (1+h_0+s+t)\Gamma (h_{-1}-h_0-s-t)}\mathrm{d}s\mathrm{d}t.$$

$$h_{-1}-h_0\in \mathbb{Z},h_0-h_1-h_2-h_3\in \mathbb{Z}\text{and}{h}_0-h_4-h_5-h_6\in \mathbb{Z},$$

$$F_6^{\prime }(h_0,h_{-1};h_1,h_2,h_3,h_4,h_5,h_6)$$

$$=\frac{(-1{)}^{(h_{-1}-h_0)+(h_0-h_1-h_2-h_3)+(h_0-h_4-h_5-h_6)}\Gamma (h_{-1})}{\Gamma (1+h_0-h_1-h_2)\Gamma (1+h_0-h_1-h_3)\Gamma (1+h_0-h_2-h_3)}$$

$$\times \frac{1}{\Gamma (1+h_0-h_4-h_5)\Gamma (1+h_0-h_4-h_6)\Gamma (1+h_0-h_5-h_6)}$$

$$\displaystyle\quad\quad\times\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma(h_{4}+t)\,\Gamma(h_{5}+t)\,\Gamma(h_{6}+t)}{\Gamma(1+t)\,\Gamma(1+h_{0}-h_{-1}+t)\,\Gamma(h_{4}+h_{5}+h_{6}-h_{0}+t)}\,\biggl{(}\frac{\pi}{\sin\pi t}\biggr{)}^{3}$$

$$\displaystyle\quad\quad\;\times\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma(h_{1}+s)\,\Gamma(h_{2}+s)\,\Gamma(h_{3}+s)}{\Gamma(1+s)\,\Gamma(1+h_{0}-h_{-1}+s)\,\Gamma(h_{1}+h_{2}+h_{3}-h_{0}+s)}\,\biggl{(}\frac{\pi}{\sin\pi s}\biggr{)}^{3}$$

$$\times \frac{\Gamma (1+h_0-h_{-1}+s+t)}{\Gamma (1+h_0+s+t)}\frac{\sin \pi (s+t)}{\pi }\mathrm{d}s\mathrm{d}t.$$

$$\displaystyle F_{6}^{\operatorname{sym}}(n)=F_{6}^{\prime}(3n+2,3n+2;n+1,\dots,n+1)=\frac{1}{2\pi i}\int_{c_{1}-i\infty}^{c_{1}+i\infty}\biggl{(}\frac{(t+1)_{n}}{n!}\biggr{)}^{3}\,\biggl{(}\frac{\pi}{\sin\pi t}\biggr{)}^{3}$$

$$\displaystyle\quad\quad\;\times\frac{1}{2\pi i}\int_{c_{2}-i\infty}^{c_{2}+i\infty}\biggl{(}\frac{(s+1)_{n}}{n!}\biggr{)}^{3}\,\biggl{(}\frac{\pi}{\sin\pi s}\biggr{)}^{3}\frac{(3n+1)!}{(s+t+1)_{3n+2}}\,\frac{\sin\pi(s+t)}{\pi}\,{\mathrm{d}}s\,{\mathrm{d}}t.$$

$$\frac{(3n+1)!}{(s+t+1{)}_{3n+2}}\frac{\sin \pi (s+t)}{\pi }$$

$$\biggl{(}\frac{(s+1)_{n}}{n!}\biggr{)}^{3}\,\biggl{(}\frac{\pi}{\sin\pi s}\biggr{)}^{3}$$

$$\biggl{(}\frac{(t+1)_{n}}{n!}\biggr{)}^{3}\,\biggl{(}\frac{\pi}{\sin\pi t}\biggr{)}^{3}.$$

$$\sin \pi (s+t)=\sin \pi s\cos \pi t+\cos \pi s\sin \pi t,$$

$$\frac{1}{2}{F}_6^{\mathrm{sym}}(n)$$

$$\displaystyle\quad\times\frac{1}{2\pi i}\int_{c-n-i\infty}^{c-n+i\infty}\biggl{(}\frac{(s+1)_{n}}{n!}\biggr{)}^{3}\,\frac{(3n+1)!}{(s+t+1)_{3n+2}}\biggl{(}\frac{\pi}{\sin\pi s}\biggr{)}^{2}{\mathrm{d}}s\,{\mathrm{d}}t$$

$$\displaystyle\biggl{(}\frac{(s+1)_{n}}{n!}\biggr{)}^{3}\,\frac{(3n+1)!}{(s+t+1)_{3n+2}}=\sum_{k=1}^{3n+2}\frac{A_{k}(t)}{s+t+k},$$

$$\displaystyle\text{where}\quad A_{k}(t)=(-1)^{k-1}\binom{3n+1}{k-1}\biggl{(}\frac{(-t-k+1)_{n}}{n!}\biggr{)}^{3}\quad\text{for}\;k=1,2,\dots,3n+2.$$

$$H_n(t)$$

$$\displaystyle=-\sum_{\nu=-n}^{\infty}\frac{\partial}{\partial s}\biggl{(}\biggl{(}\frac{(s+1)_{n}}{n!}\biggr{)}^{3}\frac{(3n+1)!}{(s+t+1)_{3n+2}}\biggr{)}\bigg{|}_{s=\nu}$$

Full text

1. Introduction

Apéry’s proof [1, 6, 18] of the irrationality of $\zeta(3)$ in the 1970s sparked research in arithmetic of the values of Riemann’s zeta function $\zeta(s)$ at integers $s\geq 2$. Some particular representatives of this development include [4, 9, 8, 13], and the story culminated in a remarkable arithmetic method [14, 15] of Rhin and Viola to produce sharp irrationality measures for $\zeta(2)$ and $\zeta(3)$ using groups of transformations of rational approximations to the quantities. In spite of hopes to (promptly) extend Apéry’s success to $\zeta(5)$ and other zeta values, the next achievement in this direction [3, 16] materialised only in the 2000s in the work of Ball and Rivoal. The latter result helped to unify differently looking approaches for arithmetic investigations of zeta values $\zeta(s)$ 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 $\zeta(4)$, 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 are not good enough to conclude about its irrationality. In [19], a general construction of rational approximations to $\zeta(4)$ is proposed, which makes use of very-well-poised hypergeometric integrals and a group of their transformations; it 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 as ‘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 $\zeta(4)$ from [19] in a different form (still hypergeometric!) and obtain, by these means, a better control of the arithmetic of their coefficients. On this way we are able to produce the estimate

[TABLE]

for the irrationality exponent of the zeta value, which is better than the conjectural one announced in [19]. This is not surprising, as we do not attempt at proving the denominator conjecture from [19] but instead investigate the arithmetic of approximations from the different hypergeometric family.

The plan of our exposition below is as follows. In Section 2 we give a Barnes-type double integral for rational approximations to $\zeta(4)$ and then, in Section 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 Section 4 and embed the approximations into a 12-parametric family of hypergeometric-type sums that are further discussed in greater details in Section 5. Furthermore, Section 6 reviews (and recovers) the permutation group related to the linear forms in $1$ and $\zeta(4)$ from a special subfamily of the approximations constructed. Finally, we investigate arithmetic aspects of the general rational approximations in Section 7 and produce a calculation that leads to the new bound for $\mu(\zeta(4))$ in Section 8.

In the text below, we intentionally avoid producing claims (in the form of propositions and lemmas) to make our exposition a storytelling rather than a traditional mathematical writing.

2. Integral representations

For $k\geq 2$ even, fix a generic set of complex parameters

[TABLE]

satisfying the conditions

[TABLE]

and define as in [21] the very-well-poised hypergeometric integrals

[TABLE]

By Bailey’s integral analogue of Dougall’s theorem [2, Section 6.6],

[TABLE]

Substituting this into the iteration

[TABLE]

obtained in [21, Section 3] we deduce consequently that

[TABLE]

and

[TABLE]

Furthermore, if

[TABLE]

then the latter can be given as

[TABLE]

3. The most symmetric case

Equation (1) has an interesting structure. For example, in the most symmetric case it implies

[TABLE]

Notice that the function

[TABLE]

is entire in both its variables, while the poles of

[TABLE]

in a right half-plane are at $s=0,1,2,\dots$; and the latter function is analytic in the strip $-(n+1)<\operatorname{Re}s<0$. A similar structure is for

[TABLE]

This implies that one can take $c_{1},c_{2}\in\mathbb{R}$ to be any in the range $-(n+1)<c_{1},c_{2}<0$; we choose $c_{1}=c_{2}=c-n$ with $c=-1/3$ for our discussion below.

Now write

[TABLE]

so that the integral is split into the integration

[TABLE]

(twice, because of the symmetry $s\leftrightarrow t$).

We first deal with the internal integral in (2). The rational integrand is decomposed into the sum of partial fractions:

[TABLE]

Then

[TABLE]

because

[TABLE]

by the residue sum theorem. The choices $\nu_{0}=0$ and $\nu_{0}=-n$ lead to the equality

[TABLE]

since $A_{k}(t)$ are polynomials, the two representations imply that the only poles of $H_{n}(t)$ are located at the integers

[TABLE]

Furthermore, the function

[TABLE]

has only poles at $t=-(n+1),-(n+2),\dots,-(2n+1)$ and vanishes at $t=-1,-2,\dots,-n$. Moreover, $\widetilde{H}_{n}(t)$ 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

[TABLE]

each of degree at most $-2$ (in $t$). This means that we have a partial-fraction decomposition

[TABLE]

with $\sum_{j=n+1}^{2n+1}C_{j}=0$. With the help of the following consequence of formula (3),

[TABLE]

we find out that

[TABLE]

Note that

[TABLE]

and

[TABLE]

for all $j,k\in\mathbb{Z}$ by the standard arithmetic properties of integer-valued polynomials [22, Lemma 4], where $d_{n}$ denotes the least common multiple of $1,2,\dots,n$. Furthermore, each term of the sums for $B_{j}$ and $C_{j}$ has a factor of the form

[TABLE]

and these quantities are all divisible by the greatest common divisor $\Phi_{n}$ of numbers

[TABLE]

(there are only finitely many nonzero products on the list). Thus,

[TABLE]

Now

[TABLE]

Since

[TABLE]

we have

[TABLE]

This implies that

[TABLE]

In Section 7 we reveal details of computing $\Phi_{n}$ (and its asymptotics as $n\to\infty$); we show that $\Phi_{n}$ is divisible by the product over primes

[TABLE]

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.

4. (

Old approximations to \003\266(4))Old approximations to $\zeta(4)$

We now concentrate on a specific setting of Section 2: $k=6$ and the parameters

[TABLE]

are positive integers satisfying the conditions

[TABLE]

Define the rational function

[TABLE]

with

[TABLE]

Then

[TABLE]

is essentially the very-well-poised hypergeometric integral given in [19]; notice, however, that the arithmetic normalisation factor $\gamma(\boldsymbol{h})$ slightly differs from the one used in [19]. Rearranging the order of parameters in (1) we obtain

[TABLE]

The double integral we arrive at belongs to a more general (12-parametric) family, which we are going to discuss in the next section.

5. (

General approximations to \003\266(4))General approximations to $\zeta(4)$

The integral in (5) is a special case of

[TABLE]

where the integral parameters

[TABLE]

are subject to the conditions

[TABLE]

and

[TABLE]

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 does not affect $G(\boldsymbol{a},\boldsymbol{b})$; 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 to assume that $b_{1}=b_{2}=1$.) The latter two symmetries potentially leave 12 out of 14 parameters (7) independent. Furthermore, we choose

[TABLE]

to be a reordering of the parameters (7) (so that (9) and (7) coincide as multi-sets) such that

[TABLE]

Additionally, we assume

[TABLE]

Similarly to the most symmetric case in Section 3, we may choose the integration paths in (6) to be the vertical lines $\{c_{1}+iy:y\in\mathbb{R}\}$ for $s$ and $\{c_{2}+iy:y\in\mathbb{R}\}$ for $t$, with

[TABLE]

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

[TABLE]

are bounded in their respective integration domains. By

[TABLE]

the integral $G(\boldsymbol{a},\boldsymbol{b})$ 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

[TABLE]

As already seen in the most symmetric case, the integral

[TABLE]

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 $\sin\pi s$ 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 $(t+s+a_{0})_{b_{0}-a_{0}}$, 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

[TABLE]

where

[TABLE]

satisfy $\sum_{k=a_{0}}^{b_{0}-1}A_{k}(t)=0$. Then

[TABLE]

where $\nu_{0}$ is any integer in the interval $1-a_{3}^{*}\leqslant\nu_{0}\leqslant 1-b_{3}^{*}$. Since all $A_{k}(t)$ are polynomials, the poles of function (14) are only possible at

[TABLE]

For a similar reason, with $\nu_{0}$ in the larger interval $1-a_{5}^{*}\leqslant\nu_{0}\leqslant 1-b_{1}^{*}$, the function

[TABLE]

has only poles possible at

[TABLE]

Since

[TABLE]

when $1-a_{3}^{*}\leqslant\nu_{0}\leqslant 1-b_{3}^{*}$, 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 $[a_{5}^{*}-b_{0}+1,b_{1}^{*}-a_{0}-1]$; however, $H(t)$ may still possess simple poles at integers in $[a_{3}^{*}-b_{0}+1,b_{3}^{*}-a_{0}-1]$. Arguing as in Section 3 we arrive at the partial-fraction decomposition

[TABLE]

because the rational function $\widetilde{H}(t)$ has degree at most $-2$ by (8). Noticing that the expression

[TABLE]

has at least simple zeroes at $t=1-a_{6}^{*},\dots,-b_{2}^{*}$ and at least double zeroes at $t=1-a_{4}^{*},2-a_{4}^{*},\dots,-b_{4}^{*}$ and taking into account condition (10), we find out that $\widetilde{H}(t)$ does not have poles in the half-plane $\operatorname{Re}t>c_{2}$, hence the expansion (15) ‘shortens’ to

[TABLE]

In fact, the second sum is over the interval $\max\{a_{4}^{*},1+a_{0}-b_{3}^{*}\}\leqslant j\leqslant b_{0}-a_{3}^{*}-1$, while the first one is over $\max\{a_{6}^{*},1+a_{0}-b_{1}^{*}\}\leqslant j\leqslant b_{0}-a_{5}^{*}-1$ and may be even empty if the interval is empty. With the explicit expressions (13) and (14) (used, for example, with $\nu_{0}=1-a_{3}^{*}$) in mind, we conclude that the coefficients

[TABLE]

satisfy

[TABLE]

but also

[TABLE]

for primes $p>\sqrt{b_{0}-a_{0}}$ (see [20, Lemmas 17, 18]). Furthermore,

[TABLE]

where $\sum_{j}C_{j}=0$ was implemented. Performing the same way for the second double integral in (11) we conclude that

[TABLE]

and

[TABLE]

for primes $p>\sqrt{b_{0}-a_{0}-2}$.

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 be potentially dropped without significant arithmetic losses. For example, if $b_{1}^{*}+b_{2}^{*}>a_{0}-1$ then the partial-fraction decomposition (15) translates into

[TABLE]

so that there are poles of $\widetilde{H}(t)$ to the right of the contour $\operatorname{Re}t=c_{2}$. The corresponding residues of the integrand are

[TABLE]

where $j$ is an integer in the interval $1+a_{0}-b_{3}^{*}\leqslant j\leqslant b_{4}^{*}-1$ and we use the expansion

[TABLE]

Proceeding as above we deduce that

[TABLE]

which is again seen to be a linear form in $\mathbb{Z}\zeta(4)+\mathbb{Q}$.

6. (

The group structure for \003\266(4))The group structure for $\zeta(4)$

Following [19], to any set of parameters $\boldsymbol{h}$ from Section 4 we assign the 27-element multiset of nonnegative integers

[TABLE]

and set $H(\boldsymbol{e})=F(\boldsymbol{h})$ for the quantity defined in that section. By the construction,

[TABLE]

is invariant under any permutation of the parameters $h_{1},h_{2},\dots,h_{6}$ (which we can view as the ‘$\boldsymbol{h}$-trivial’ action). Clearly, any such permutation induces the corresponding permutation of the parameter set (18).

On the other hand, it is seen from (6) that the quantity

[TABLE]

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 ‘$\boldsymbol{a}$-trivial’. (The same effect is produced by ‘$\boldsymbol{b}$-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

[TABLE]

which reflects the symmetry $s\leftrightarrow t$ of the double integral (6). In addition, we recall that $G(\boldsymbol{a},\boldsymbol{b})$ 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 ‘$(\boldsymbol{a},\boldsymbol{b})$-trivial’ action.

By setting

[TABLE]

we have $F(\boldsymbol{h})=G(\boldsymbol{a},\boldsymbol{b})$. If we request the condition

[TABLE]

to hold, then the shift of $h_{0},h_{1},h_{3},h_{5}$ by $1+h_{0}-h_{1}-h_{3}-h_{5}$, that is, the transformation

[TABLE]

induces the composition of the shift of $a_{0},a_{1},a_{3},a_{5}$ and $b_{0},b_{1},b_{3},b_{5}$ by $1+h_{0}-h_{1}-h_{3}-h_{5}$ and the permutation $(b_{1}\ b_{3})(b_{4}\ b_{6})$. Therefore $\mathfrak{b}_{135}$, which also induces the permutation

[TABLE]

on the parameter set (18), is an $(\boldsymbol{a},\boldsymbol{b})$-trivial transformation. As a consequence, the quantity

[TABLE]

does not change by the action of the permutation $\mathfrak{b}$. We remark that (20) is a very natural condition for the application of the $(\boldsymbol{a},\boldsymbol{b})$-trivial action to $F(\boldsymbol{h})$. Indeed, by (19) we have $b_{1}-b_{5}=b_{2}-b_{6}$, and (20) is equivalent to $b_{1}-b_{3}=b_{4}-b_{6}$, or to $b_{3}-b_{5}=b_{2}-b_{4}$.

Taking the multiset

[TABLE]

we conclude that the quantity

[TABLE]

is invariant under the $\boldsymbol{h}$-trivial permutations and, by

[TABLE]

also under the permutation $\mathfrak{b}$.

The permutation group of the multiset (18), which is generated by all $\boldsymbol{h}$-trivial and the permutation $\mathfrak{b}$, coincides with the group ${\mathfrak{G}}$ (of order $51840$) considered in [19]. (Note that the group contains the above involution $\mathfrak{i}$ as well.) By these means we also recover the invariance of the quantity

[TABLE]

under the action of ${\mathfrak{G}}$ and corresponding to the arithmetic normalisation of $H(\boldsymbol{e})=F(\boldsymbol{h})=G(\boldsymbol{a},\boldsymbol{b})$ in Section 4.

Because our access to the arithmetic of coefficients of linear forms $H(\boldsymbol{e})\in\mathbb{Z}\zeta(4)+\mathbb{Q}$ is performed through their $G(\boldsymbol{a},\boldsymbol{b})$-representation, we will be interested in collecting a set of representatives which are distinct modulo $(\boldsymbol{a},\boldsymbol{b})$-trivial transformations. For a generic set of integral parameters $\boldsymbol{h}$ subject to (20), such set of representatives contains 120 different elements. Indeed, by (19) and (20) the subgroup of all the $(\boldsymbol{a},\boldsymbol{b})$-trivial permutations in ${\mathfrak{G}}$ contains $3!^{3}\cdot 2!=432$ elements, and is generated by:

• •

the $\boldsymbol{a}$- and $\boldsymbol{h}$-trivial permutations $(h_{1}\ h_{3})$ and $(h_{3}\ h_{5})$;

• •

the $\boldsymbol{a}$- and $\boldsymbol{h}$-trivial permutations $(h_{2}\ h_{4})$ and $(h_{4}\ h_{6})$;

• •

the $\boldsymbol{b}$-trivial permutation $(b_{1}\ b_{3})(b_{4}\ b_{6})$ (that is, by $\mathfrak{b}_{135}$); and

• •

the involution $\mathfrak{i}$ (that is, by $(h_{1}\ h_{6})(h_{2}\ h_{5})(h_{3}\ h_{4})$).

This subgroup also contains $(b_{2}\ b_{4})(b_{3}\ b_{5})$ (namely, $\mathfrak{b}_{246}=\mathfrak{i}\mathfrak{b}_{135}\mathfrak{i}$), and is isomorphic to $\mathfrak{S}_{3}^{3}\times\mathfrak{S}_{2}$. Now, the group ${\mathfrak{G}}$ is generated by $(h_{1}\ h_{3})$, $(h_{3}\ h_{5})$, $(h_{2}\ h_{4})$, $(h_{4}\ h_{6})$, $\mathfrak{b}_{135}$ and $(h_{3}\ h_{4})$. Note that $(h_{1}\ h_{3})$, $(h_{3}\ h_{5})$, $(h_{2}\ h_{4})$ and $(h_{4}\ h_{6})$ commute with $\mathfrak{b}_{135}$, while $(h_{3}\ h_{4})$ acts on $(\boldsymbol{a},\boldsymbol{b})$ by $a_{3}\leftrightarrow a_{4}$, $b_{3}\mapsto b_{3}+a_{4}-a_{3}$, $b_{4}\mapsto b_{4}+a_{3}-a_{4}$ (and leaves $a_{i},b_{i}$ unchanged for $i\not=3,4$). Hence there are exactly $|{\mathfrak{G}}|/432=120$ elements in ${\mathfrak{G}}$ that are distinct modulo the $(\boldsymbol{a},\boldsymbol{b})$-trivial subgroup, each for any simultaneous choice of a subset $\{a_{1},a_{3},a_{5}\}$ (or $\{a_{2},a_{4},a_{6}\}$) of $\{h_{1},\dots,h_{6}\}$ (among all $\binom{6}{3}=20$ such subsets) and of a permutation in the $\boldsymbol{b}$-trivial subgroup (of $3!=6$ elements) generated by $\mathfrak{b}_{135}$ and $\mathfrak{b}_{246}$.

7. Arithmetic of linear forms

In order to compute the minimum on the right-hand side of (17), we distinguish two different situations: (a) $j+l-a_{0}$ is coprime with $p$, and (b) $j+l-a_{0}$ is divisible by $p$. In case (a), we get $\lfloor(j+l-a_{0})/p\rfloor=\lfloor(j+l-a_{0}-1)/p\rfloor$, so that the minimum in (17) is greater or equal than

[TABLE]

In case (b), we have $l=-j+a_{0}+\mu p$ for some $\mu\in\mathbb{Z}$ and

[TABLE]

for primes $p>\sqrt{b_{0}-a_{0}-2}$, where the property $\lfloor\alpha+\mu\rfloor=\lfloor\alpha\rfloor+\mu$ was used, together with the passage

[TABLE]

for the fractional part $\{\alpha\}=\alpha-\lfloor\alpha\rfloor$ of a number. This means that in case (b) the minimum in (17) is equal to

[TABLE]

Combining the two cases together we conclude that

[TABLE]

for primes $p>\sqrt{b_{0}-a_{0}-2}$, where the quantities $\Omega_{1}$ and $\Omega_{2}$ are defined in (21) and (22).

When we choose

[TABLE]

for some positive set of integer directions $(\boldsymbol{\alpha},\boldsymbol{\beta})$, then computing $\Omega_{1},\Omega_{2}$ reduces to the computation of the minima $\omega_{1}^{*}(x)$ and $\omega_{2}^{*}(x)$ of functions

[TABLE]

over $y,z$ and over $y$, respectively. Indeed,

[TABLE]

in the settings above. This means that

[TABLE]

for primes $p>\sqrt{(\beta_{0}-\alpha_{0})n}$.

Notice that the functions $\omega_{1}$ and $\omega_{2}$ (hence their minima) are $1$-periodic in each variable, so it is sufficient to compute them on the intervals $[0,1)$. In the most symmetric case

[TABLE]

we already get (by droping the four non-negative terms in both $\omega_{1}$ and $\omega_{2}$)

[TABLE]

and

[TABLE]

Indeed, when $x\in[\tfrac{2}{3},1)$, the first inequality follows from

[TABLE]

if either $y<x$ or $z<x$; otherwise, $\frac{2}{3}\leqslant x\leqslant y<1$ and $\frac{2}{3}\leqslant x\leqslant z<1$ imply

[TABLE]

A proof of the second inequality, when $x\in[\frac{1}{2},1)$, makes use of $\lfloor y-x\rfloor=-1$, $\lfloor y+x\rfloor\geq 0$ if $0\leqslant y<\frac{1}{2}$, and of $\lfloor y+x\rfloor=1$, $\lfloor y-x\rfloor\leqslant 0$ if $\frac{1}{2}\leqslant y<1$. The two inequalities together mean that the quantity $\Phi_{n}$ from Section 3 is divisible by (4).

It looks quite plausible (though we do not possess any proof of this) that we always have $\omega_{2}^{*}(x)\geq\omega_{1}^{*}(x)$ except for possibly finitely many rational points on the interval $[0,1)$. (Notice that $\omega_{1}(x,y,\alpha_{0}x-y)$ coincides with $\omega_{2}(x,y)$ apart from finitely many rational lines crossing the square $[0,1)^{2}$.)

Now assume that the linear forms $G(\boldsymbol{a},\boldsymbol{b})$ originate from the forms $F(\boldsymbol{h})$ of Section 4 and condition (20) written as

[TABLE]

holds. In this case we can scale all the parameters in (18) to discuss the set $\boldsymbol{e}n$ instead, where

[TABLE]

and record the related quantities by $H(\boldsymbol{e}n)=B(\boldsymbol{e}n)\zeta(4)-C(\boldsymbol{e}n)\in\mathbb{Z}\zeta(4)+\mathbb{Q}$, where $n=0,1,2,\dots$ . The discussion above (see (24)) implies that

[TABLE]

where

[TABLE]

hence also

[TABLE]

where ${\mathfrak{g}}\boldsymbol{e}$ denotes the image of the multiset $\boldsymbol{e}$ under the action of ${\mathfrak{g}}\in{\mathfrak{G}}$. At the same time,

[TABLE]

for all ${\mathfrak{g}}\in{\mathfrak{G}}$, in view of the invariance of $H(\boldsymbol{e}n)/\Pi(\boldsymbol{e}n)$ under the action of ${\mathfrak{G}}$ (and of the irrationality of $\zeta(4)$). This implies that

[TABLE]

for primes $p>\sqrt{(\beta_{0}-\alpha_{0})n}$ and all ${\mathfrak{g}}\in{\mathfrak{G}}$, hence

[TABLE]

where

[TABLE]

The maximum can be restricted to distinct representatives modulo the group of $(\boldsymbol{a},\boldsymbol{b})$-trivial permutations.

8. (

One concrete example of irrationality measure for \003\266(4))One concrete example of irrationality measure for $\zeta(4)$

In the notation of Section 4 we take

[TABLE]

with

[TABLE]

If we set $F_{n}=F(\boldsymbol{h})=G(\boldsymbol{a},\boldsymbol{b})=u_{n}\zeta(4)-v_{n}$ then the asymptotics of $F_{n}$ and $u_{n}$ as $n\to\infty$ is computed with the help of [19, Proposition 1] (adapted here to address a a slightly different normalisation of $F(\boldsymbol{h})$):

[TABLE]

The above choice of $\boldsymbol{h}$ translates the form $F_{n}=F(\boldsymbol{h})$ from (5) into $G(\boldsymbol{a},\boldsymbol{b})$ from (6) with the parameters (23) as follows:

[TABLE]

The denominator of $v_{n}=C(\boldsymbol{a},\boldsymbol{b})$ in (16) is $d_{21n}^{3}d_{23n}^{\vphantom{1}}$. The following table lists 31 out of 120 representatives under the action of group ${\mathfrak{G}}$ on (26) modulo the trivial $(\boldsymbol{a},\boldsymbol{b})$-action, only those that contribute to the computation of the corresponding function $\omega(x)=\omega(\boldsymbol{e};x)$ in (25):

[TABLE]

[TABLE]

[TABLE]

Here we give the representatives in the format $(\beta_{0};\alpha_{1},\dots,\alpha_{6})=(\eta_{0};\eta_{1},\dots,\eta_{6})$; all other parameters are completely determined by the data.

Then

[TABLE]

where the notation $[a,b)^{\langle N\rangle}$ means that the maximum in (25) is attained on the $N$-th representative. Denoting

[TABLE]

we conclude that $\Phi_{n}^{-1}u_{n}\in\mathbb{Z}$ and $\Phi_{n}^{-1}d_{21n}^{3}d_{23n}v_{n}\in\mathbb{Z}$; in other words,

[TABLE]

At the same time the asymptotics of $\Phi_{n}^{-1}d_{21n}^{3}d_{23n}$ is controlled by the prime number theorem:

[TABLE]

where $\psi(x)$ is the logarithmic derivative of the gamma function. Now [19, Proposition 3] applies to imply that the irrationality exponent of $\zeta(4)$ is bounded above by

[TABLE]

Finally, we point out that the general family of rational approximations to $\zeta(4)$ from Section 5 is only exploited here when it is linked to the old approximations reviewed in Section 4. A reason behind this is mainly an easy access to the asymptotic behaviour of the corresponding forms $G(\boldsymbol{a},\boldsymbol{b})$ and their coefficients $B(\boldsymbol{a},\boldsymbol{b})$. One may hope to get a better control of general approximations from Section 5 by covering analytic aspects of the 12-parametric family there, however this will not necessarily lead to (significantly) better arithmetic consequences.