Hypergeometric rational approximations to ζ(4)
aa r X i v : . [ m a t h . N T ] M a y HYPERGEOMETRIC RATIONAL APPROXIMATIONS TO ζ (4) RAFFAELE MARCOVECCHIO AND WADIM ZUDILIN
Abstract.
We give a new hypergeometric construction of rational approxima-tions to ζ (4), which absorbs the earlier one from 2003 based on Bailey’s F hypergeometric integrals. With the novel ingredients we are able to get a bettercontrol of arithmetic and produce a record irrationality measure for ζ (4). Introduction
Ap´ery’s proof [1, 6, 18] of the irrationality of ζ (3) in the 1970s sparked researchin arithmetic of the values of Riemann’s zeta function ζ ( s ) at integers s ≥
2. Someparticular representatives of this development include [4, 9, 8, 13], and the storyculminated in a remarkable arithmetic method [14, 15] of Rhin and Viola to pro-duce sharp irrationality measures for ζ (2) and ζ (3) using groups of transformationsof rational approximations to the quantities. In spite of hopes to (promptly) extendAp´ery’s success to ζ (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 latterresult helped to unify differently looking approaches for arithmetic investigationsof zeta values ζ ( s ) and related constants under a ‘hypergeometric’ umbrella, withsome particular highlights given in [19, 20] by one of these authors. The hypergeo-metric machinery has proven to be useful in further arithmetic applications; see, forexample, [7, 11, 12, 22] for more recent achievements.The quantity ζ (4), though known to be irrational and even transcendental, re-mains a natural target for testing the hypergeometry. Ap´ery-type approximationsto 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 gen-eral construction of rational approximations to ζ (4) is proposed, which makes useof very-well-poised hypergeometric integrals and a group of their transformations;it leads to an estimate for the irrationality exponent of the number in questionprovided that a certain ‘denominator conjecture’ for the rational approximationsis valid. The conjecture appears to be difficult enough, with its only special caseestablished in [10] but insufficient for arithmetic applications. This case is usuallydubbed as ‘most symmetric’, because the group of transformations acts trivially onthe corresponding approximations.The principal goal of this work is to recast the rational approximations to ζ (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 Date : May 2019.2010
Mathematics Subject Classification.
Primary 11J82; Secondary 11Y60, 33C20, 33C60. produce the estimate µ ( ζ (4)) . . . . for the irrationality exponent of the zeta value, which is better than the conjecturalone announced in [19]. This is not surprising, as we do not attempt at proving thedenominator conjecture from [19] but instead investigate the arithmetic of approxi-mations from the different hypergeometric family.The plan of our exposition below is as follows. In Section 2 we give a Barnes-typedouble integral for rational approximations to ζ (4) and then, in Section 3, workout the particular ‘most symmetric’ case of this integral, which clearly illustratesarithmetic features of the new representation of the approximations. We recallgeneral settings from [19] in Section 4 and embed the approximations into a 12-parametric family of hypergeometric-type sums that are further discussed in greaterdetails in Section 5. Furthermore, Section 6 reviews (and recovers) the permutationgroup related to the linear forms in 1 and ζ (4) from a special subfamily of theapproximations constructed. Finally, we investigate arithmetic aspects of the generalrational approximations in Section 7 and produce a calculation that leads to the newbound for µ ( ζ (4)) in Section 8.In the text below, we intentionally avoid producing claims (in the form of propo-sitions and lemmas) to make our exposition a storytelling rather than a traditionalmathematical writing. 2. Integral representations
For k ≥ generic set of complex parameters h = ( h , h − ; h , h , . . . , h k )satisfying the conditionsmax { , Re( h − h − ) } < Re h j <
12 Re h for j = 1 , . . . , k, and define as in [21] the very-well-poised hypergeometric integrals F ′ k ( h ) = F ′ k ( h , h − ; h , h , . . . , h k )= 12 πi Z i ∞− i ∞ ( h + 2 t ) Q kj = − Γ( h j + t ) · Γ( h − − h − t ) Γ( − t ) Q kj =1 Γ(1 + h − h j + t ) d t. By Bailey’s integral analogue of Dougall’s theorem [2, Section 6.6], F ′ ( h , h − ; h , h ) = Γ( h − ) Γ( h ) Γ( h ) Γ( h + h − − h ) Γ( h + h − − h )Γ(1 + h − h − h ) Γ( h + h + h − − h ) . YPERGEOMETRIC RATIONAL APPROXIMATIONS TO ζ (4) 3 Substituting this into the iteration F ′ k +2 ( h , h − ; h , . . . , h k − , h k , h k +1 , h k +2 )= 1Γ(1 + h − h k − h k +1 ) Γ(1 + h − h k − h k +2 ) Γ(1 + h − h k +1 − h k +2 ) × πi Z i ∞− i ∞ Γ( h k + s ) Γ( h k +1 + s ) Γ( h k +2 + s ) × Γ(1 + h − h k − h k +1 − h k +2 − s ) · F ′ k ( h , h − ; − s, h , . . . , h k − ) d s obtained in [21, Section 3] we deduce consequently that F ′ ( h , h − ; h , h , h , h )= Γ( h − ) Γ( h ) Γ( h + h − − h )Γ(1 + h − h − h ) Γ(1 + h − h − h ) Γ(1 + h − h − h ) × πi Z i ∞− i ∞ Γ( h + s ) Γ( h + s ) Γ( h + s ) Γ(1 + h − h − h − h − s ) × Γ( h − − h − s ) Γ( − s )Γ(1 + h − h + s ) Γ( h + h − − h − s ) d s and F ′ ( h , h − ; h , h , h , h , h , h )= 1Γ(1 + h − h − h ) Γ(1 + h − h − h ) Γ(1 + h − h − h ) × πi Z i ∞− i ∞ Γ( h + t ) Γ( h + t ) Γ( h + t ) × Γ(1 + h − h − h − h − t ) · F ′ ( h , h − ; − t, h , h , h ) d t = 1Γ(1 + h − h − h ) Γ(1 + h − h − h ) Γ(1 + h − h − h ) × Γ( h − )Γ(1 + h − h − h ) Γ(1 + h − h − h ) Γ(1 + h − h − h ) × πi Z i ∞− i ∞ Γ( h + t ) Γ( h + t ) Γ( h + t ) Γ(1 + h − h − h − h − t ) × πi Z i ∞− i ∞ Γ( h + s ) Γ( h + s ) Γ( h + s ) Γ(1 + h − h − h − h − s ) × Γ( h − − h − t ) Γ( − t ) Γ( h − − h − s ) Γ( − s )Γ(1 + h + s + t ) Γ( h − − h − s − t ) d s d t. Furthermore, if h − − h ∈ Z , h − h − h − h ∈ Z and h − h − h − h ∈ Z , RAFFAELE MARCOVECCHIO AND WADIM ZUDILIN then the latter can be given as F ′ ( h , h − ; h , h , h , h , h , h )= ( − ( h − − h )+( h − h − h − h )+( h − h − h − h ) Γ( h − )Γ(1 + h − h − h ) Γ(1 + h − h − h ) Γ(1 + h − h − h ) × h − h − h ) Γ(1 + h − h − h ) Γ(1 + h − h − h ) × πi Z i ∞− i ∞ Γ( h + t ) Γ( h + t ) Γ( h + t )Γ(1 + t ) Γ(1 + h − h − + t ) Γ( h + h + h − h + t ) (cid:18) π sin πt (cid:19) × πi Z i ∞− i ∞ Γ( h + s ) Γ( h + s ) Γ( h + s )Γ(1 + s ) Γ(1 + h − h − + s ) Γ( h + h + h − h + s ) (cid:18) π sin πs (cid:19) × Γ(1 + h − h − + s + t )Γ(1 + h + s + t ) sin π ( s + t ) π d s d t. (1)3. The most symmetric case
Equation (1) has an interesting structure. For example, in the most symmetriccase it implies F sym6 ( n ) = F ′ (3 n + 2 , n + 2; n + 1 , . . . , n + 1) = 12 πi Z c + i ∞ c − i ∞ (cid:18) ( t + 1) n n ! (cid:19) (cid:18) π sin πt (cid:19) × πi Z c + i ∞ c − i ∞ (cid:18) ( s + 1) n n ! (cid:19) (cid:18) π sin πs (cid:19) (3 n + 1)!( s + t + 1) n +2 sin π ( s + t ) π d s d t. Notice that the function (3 n + 1)!( s + t + 1) n +2 sin π ( s + t ) π is entire in both its variables, while the poles of (cid:18) ( s + 1) n n ! (cid:19) (cid:18) π sin πs (cid:19) in a right half-plane are at s = 0 , , , . . . ; and the latter function is analytic in thestrip − ( n + 1) < Re s <
0. A similar structure is for (cid:18) ( t + 1) n n ! (cid:19) (cid:18) π sin πt (cid:19) . This implies that one can take c , c ∈ R to be any in the range − ( n +1) < c , c < c = c = c − n with c = − / π ( s + t ) = sin πs cos πt + cos πs sin πt, YPERGEOMETRIC RATIONAL APPROXIMATIONS TO ζ (4) 5 so that the integral is split into the integration12 F sym6 ( n ) = 12 πi Z c − n + i ∞ c − n − i ∞ (cid:18) ( t + 1) n n ! (cid:19) (cid:18) π sin πt (cid:19) cos πt × πi Z c − n + i ∞ c − n − i ∞ (cid:18) ( s + 1) n n ! (cid:19) (3 n + 1)!( s + t + 1) n +2 (cid:18) π sin πs (cid:19) d s d t (2)(twice, because of the symmetry s ↔ t ).We first deal with the internal integral in (2). The rational integrand is decom-posed into the sum of partial fractions: (cid:18) ( s + 1) n n ! (cid:19) (3 n + 1)!( s + t + 1) n +2 = n +2 X k =1 A k ( t ) s + t + k , where A k ( t ) = ( − k − (cid:18) n + 1 k − (cid:19)(cid:18) ( − t − k + 1) n n ! (cid:19) for k = 1 , , . . . , n + 2 . Then H n ( t ) = 12 πi Z c − n + i ∞ c − n − i ∞ (cid:18) ( s + 1) n n ! (cid:19) (3 n + 1)!( s + t + 1) n +2 (cid:18) π sin πs (cid:19) d s = − ∞ X ν = − n ∂∂s (cid:18)(cid:18) ( s + 1) n n ! (cid:19) (3 n + 1)!( s + t + 1) n +2 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) s = ν (take ν any from the interval − n ν − ∞ X ν = ν ∂∂s (cid:18)(cid:18) ( s + 1) n n ! (cid:19) (3 n + 1)!( s + t + 1) n +2 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) s = ν = ∞ X ν = ν n +2 X k =1 A k ( t )( ν + t + k ) = n +2 X k =1 A k ( t ) ∞ X ν = ν ν + t + k ) = n +2 X k =1 A k ( t ) (cid:18) ∞ X l =1 − k − X l =1 (cid:19) t + l + ν ) = − n +2 X k =1 A k ( t ) k − X l =1 t + l + ν ) , because n +2 X k =1 A k ( t ) = 0by the residue sum theorem. The choices ν = 0 and ν = − n lead to the equality − n +2 X k =1 A k ( t ) k − X l =1 t + l ) = H n ( t ) = − n +2 X k =1 A k ( t ) k − X l =1 t + l − n ) ; (3) RAFFAELE MARCOVECCHIO AND WADIM ZUDILIN since A k ( t ) are polynomials, the two representations imply that the only poles of H n ( t ) are located at the integers {− , − , . . . , − n, − (3 n + 1) } ∩ {− (2 n + 1) , − n, . . . , n − , n − } = {− , − , . . . , − n, − (2 n + 1) } . Furthermore, the function e H n ( t ) = (cid:18) ( t + 1) n n ! (cid:19) H n ( t )has only poles at t = − ( n + 1) , − ( n + 2) , . . . , − (2 n + 1) and vanishes at t = − , − , . . . , − n . Moreover, e H n ( t ) is in fact a rational function of degree at most − (cid:18) ( t + 1) n n ! (cid:19) ∂∂s (cid:18)(cid:18) ( s + 1) n n ! (cid:19) (3 n + 1)!( s + t + 1) n +2 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) s = ν , where ν = 0 , , , . . . , each of degree at most − t ). This means that we have a partial-fraction de-composition e H n ( t ) = n +1 X j = n +1 (cid:18) B j ( t + j ) + C j t + j (cid:19) with P n +1 j = n +1 C j = 0. With the help of the following consequence of formula (3), H n ( t ) = − n +1 X l =1 t + l − n ) n +2 X k = l +1 A k ( t ) = − n +1 X j = − n +1 t + j ) n +2 X k = j + n +1 A k ( t ) , we find out that B j = e H n ( t )( t + j ) (cid:12)(cid:12) t = − j = − (cid:18) ( − j + 1) n n ! (cid:19) n +2 X k = j + n +1 A k ( − j )= (cid:18) ( − j + 1) n n ! (cid:19) n +2 X k = j + n +1 ( − k (cid:18) n + 1 k − (cid:19)(cid:18) ( j − k + 1) n n ! (cid:19) and similarly C j = ∂∂t (cid:0) e H n ( t )( t + j ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) t = − j = ∂∂t (cid:18) ( − t + 1) n n ! (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) t = − j · n +2 X k = j + n +1 ( − k (cid:18) n + 1 k − (cid:19)(cid:18) ( j − k + 1) n n ! (cid:19) + (cid:18) ( − j + 1) n n ! (cid:19) n +2 X k = j + n +1 ( − k (cid:18) n + 1 k − (cid:19) · ∂∂t (cid:18) ( − t − k + 1) n n ! (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) t = − j . YPERGEOMETRIC RATIONAL APPROXIMATIONS TO ζ (4) 7 Note that ( − j + 1) n n ! ∈ Z , ( j − k + 1) n n ! ∈ Z and d n · ∂∂t (cid:18) ( − t + 1) n n ! (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t = − j ∈ Z , d n · ∂∂t (cid:18) ( − t − k + 1) n n ! (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t = − j ∈ Z 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 , , . . . , n . Further-more, each term of the sums for B j and C j has a factor of the form( − j + 1) n n ! (cid:18) n + 1 k − (cid:19) ( j − k + 1) n n ! = (cid:18) n + 1 k − (cid:19)(cid:18) j − n (cid:19)(cid:18) k − j − n (cid:19) , and these quantities are all divisible by the greatest common divisor Φ n of numbers (cid:18) n + 1 a + b + 1 (cid:19)(cid:18) an (cid:19)(cid:18) bn (cid:19) , where a, b ∈ Z (there are only finitely many nonzero products on the list). Thus,Φ − n B j ∈ Z and Φ − n d n C j ∈ Z for j = n + 1 , . . . , n + 1 . Now 12 F sym6 ( n ) = 12 πi Z c − n + i ∞ c − n − i ∞ e H n ( t ) (cid:18) π sin πt (cid:19) cos πt d t. Since (cid:18) π sin πt (cid:19) cos πt = 1( t − ν ) + O ( t − ν ) as t → ν ∈ Z , we have12 F sym6 ( n ) = ∞ X ν = − n Res t = ν e H n ( t ) (cid:18) π sin πt (cid:19) cos πt = 12 ∞ X ν = − n ∂ e H n ( t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) t = ν = 12 ∞ X ν = − n n +1 X j = n +1 (cid:18) B j ( ν + j ) + 2 C j ( ν + j ) (cid:19) = 3 n +1 X j = n +1 B j ∞ X ν = − n ν + j ) + n +1 X j = n +1 C j ∞ X ν = − n ν + j ) = 3 n +1 X j = n +1 B j · ζ (4) − (cid:18) n +1 X j = n +1 B j j − n − X l =1 l + n +1 X j = n +1 C j j − n − X l =1 l (cid:19) . This implies that 12 Φ − n d n F sym6 ( n ) ∈ Z ζ (4) + Z . RAFFAELE MARCOVECCHIO AND WADIM ZUDILIN
In Section 7 we reveal details of computing Φ n (and its asymptotics as n → ∞ ); weshow that Φ n is divisible by the product over primes Y p> √ n { n/p } < p. (4)This corresponds to the ‘denominator conjecture’ from [19]; for the most symmetriccase in this section it was established earlier in [10] using different hypergeometrictechniques. 4. Old approximations to ζ (4)We now concentrate on a specific setting of Section 2: k = 6 and the parameters h = ( h , h − ; h , h , h , h , h , h )are positive integers satisfying the conditions h − h − < h j < h for j = 1 , , , , , . Define the rational function R ( t ) = R ( h ; t ) = γ ( h ) ( h + 2 t ) Q j = − Γ( h j + t ) Q j = − Γ(1 + h − h j + t )= ( h + 2 t ) ( t + 1) h − ( h − t + 1 + h − h ) h − ( h − × ( t + 1 + h − h ) h + h − − h − ( h + h − − h − t + 1 + h − h − ) h + h − − h − ( h + h − − h − × ( h − h − h )!( t + h ) h − h − h +1 ( h − h − h )!( t + h ) h − h − h +1 × ( h − h − h )!( t + h ) h − h − h +1 ( h − h − h )!( t + h ) h − h − h +1 with γ ( h ) = ( h − h − h )! ( h − h − h )! ( h − h − h )! ( h − h − h )!( h − h − h + h − − h − h + h − − h − . Then F ( h ) = γ ( h ) F ′ ( h ) = − ∞ X t = t dd t R ( h ; t ) ∈ Q + Q ζ (4)with any t ∈ Z , − min j { h j } t − max { , h − h − } , is essentially the very-well-poised hypergeometric integral given in [19]; notice, how-ever, that the arithmetic normalisation factor γ ( h ) slightly differs from the one used YPERGEOMETRIC RATIONAL APPROXIMATIONS TO ζ (4) 9 in [19]. Rearranging the order of parameters in (1) we obtain F ( h ) = ( − h − + h + h + ··· + h ( h − − γ ( h )( h − h − h )! ( h − h − h )! ( h − h − h )! × h − h − h )! ( h − h − h )! ( h − h − h )! × πi Z i ∞− i ∞ Γ( h + t ) Γ( h + t ) Γ( h + t )Γ(1 + t ) Γ(1 + h − h − + t ) Γ( h + h + h − h + t ) (cid:18) π sin πt (cid:19) × πi Z i ∞− i ∞ Γ( h + s ) Γ( h + s ) Γ( h + s )Γ(1 + s ) Γ(1 + h − h − + s ) Γ( h + h + h − h + s ) (cid:18) π sin πs (cid:19) × Γ(1 + h − h − + s + t )Γ(1 + h + s + t ) sin π ( s + t ) π d s d t = ( − h − + ··· + h × πi Z i ∞− i ∞ ( t + 1) h − ( h − t + h + h + h − h ) h − h − h ( h − h − h )! × ( t + 1 + h − h − ) h + h − − h − ( h + h − − h − (cid:18) π sin πt (cid:19) × πi Z i ∞− i ∞ ( s + 1) h − ( h − s + h + h + h − h ) h − h − h ( h − h − h )! × ( s + 1 + h − h − ) h + h − − h − ( h + h − − h − (cid:18) π sin πs (cid:19) × ( h − − t + s + 1 + h − h − ) h − sin π ( s + t ) π d s d t. (5)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 ζ (4)The integral in (5) is a special case of G ( a , b ) = 12 πi Z i ∞− i ∞ ( t + b ) a − b ( a − b )! ( t + b ) a − b ( a − b )! ( t + b ) a − b ( a − b )! (cid:18) π sin πt (cid:19) × πi Z i ∞− i ∞ ( s + b ) a − b ( a − b )! ( s + b ) a − b ( a − b )! ( s + b ) a − b ( a − b )! (cid:18) π sin πs (cid:19) × ( b − a − t + s + a ) b − a sin π ( s + t ) π d s d t, (6)where the integral parameters a = ( a ; a , a , a , a , a , a ) and b = ( b ; b , b , b , b , b , b ) (7) are subject to the conditions b − a − ≥ ( a + a + a ) − ( b + b + b ) ,b − a − ≥ ( a + a + a ) − ( b + b + b ) , (8)and max { b , b , b } min { a , a , a } , max { b , b , b } min { a , a , a } . Note that simultaneous shifts of a , a , a , a and b , b , b , b by the same integerdoes not affect G ( a , b ); the same is true for simultaneous shifts of a , a , a , a and b , b , b , b . (In particular, the shifts by given 1 − b and 1 − b , respectively, allowto assume that b = b = 1.) The latter two symmetries potentially leave 12 out of14 parameters (7) independent. Furthermore, we choose a ∗ = ( a ; a ∗ , a ∗ , a ∗ , a ∗ , a ∗ , a ∗ ) and b ∗ = ( b ; b ∗ , b ∗ , b ∗ , b ∗ , b ∗ , b ∗ ) (9)to be a reordering of the parameters (7) (so that (9) and (7) coincide as multi-sets)such that a ∗ a ∗ a ∗ , b ∗ b ∗ b ∗ and a ∗ a ∗ a ∗ , b ∗ b ∗ b ∗ . Additionally, we assume a + 1 ≥ b ∗ + b ∗ . (10)Similarly to the most symmetric case in Section 3, we may choose the integrationpaths in (6) to be the vertical lines { c + iy : y ∈ R } for s and { c + iy : y ∈ R } for t , with − a ∗ < c < − b ∗ , − a ∗ < c < − b ∗ , and we take c = 1 / − a ∗ and c = 1 / − a ∗ . Also, the rational function in s and t at the integrand in (6) has degree at most − s and in t , and the functions1sin πs , cos πs (sin πs ) and 1sin πt are bounded in their respective integration domains. Bysin π ( s + t ) = sin πs cos πt + cos πs sin πt the integral G ( a , b ) is split into two absolutely convergent integrals, and, after in-terchanging the order of integrations in s and in t in the second integral, we obtain G ( a , b ) = 12 πi Z i ∞− i ∞ ( t + b ) a − b ( a − b )! ( t + b ) a − b ( a − b )! ( t + b ) a − b ( a − b )! (cid:18) π sin πt (cid:19) cos πt × πi Z i ∞− i ∞ ( s + b ) a − b ( a − b )! ( s + b ) a − b ( a − b )! ( s + b ) a − b ( a − b )! (cid:18) π sin πs (cid:19) × ( b − a − t + s + a ) b − a d s d t + a similar integral with a j , b j changed to a − j , b − j for j = 1 , . . . , . (11) YPERGEOMETRIC RATIONAL APPROXIMATIONS TO ζ (4) 11 As already seen in the most symmetric case, the integral H ( t ) = H ( a , a , a , a ; b , b , b , b ; t )= 12 πi Z c + i ∞ c − i ∞ ( s + b ) a − b ( a − b )! ( s + b ) a − b ( a − b )! ( s + b ) a − b ( a − b )! × ( b − a − t + s + a ) b − a (cid:18) π sin πs (cid:19) d s (12)is a rational function in t , and we may even vary c in the interval − a ∗ < c < − b ∗ ,because a power of sin πs is dropped in the denominator of (12) with respect to theintegral (6). In executing this, we do not have to take care of possible poles comingfrom ( t + s + a ) b − a , because it never vanishes if t is chosen in an appropriate regionof the complex plane, and two rational functions that coincide in such a region mustcoincide everywhere.Explicitly, we have( s + b ) a − b ( a − b )! ( s + b ) a − b ( a − b )! ( s + b ) a − b ( a − b )! ( b − a − t + s + a ) b − a = b − X k = a A k ( t ) t + s + k , where A k ( t ) = ( − k + a (cid:18) b − a − k − a (cid:19) ( − t − k + b ) a − b ( a − b )! × ( − t − k + b ) a − b ( a − b )! ( − t − k + b ) a − b ( a − b )! for k = a , . . . , b − P b − k = a A k ( t ) = 0. Then H ( t ) = − ∞ X ν = ν ∂∂s b − X k = a A k ( t ) t + s + k (cid:12)(cid:12)(cid:12)(cid:12) s = ν = b − X k = a A k ( t ) ∞ X ν = ν ν + t + k ) = − b − X k = a A k ( t ) k − X l = a t + l + ν ) , (14)where ν is any integer in the interval 1 − a ∗ ν − b ∗ . Since all A k ( t ) arepolynomials, the poles of function (14) are only possible at t = a ∗ − b + 1 , a ∗ − b + 2 , . . . , b ∗ − a − . For a similar reason, with ν in the larger interval 1 − a ∗ ν − b ∗ , the function I ( t ) = ∞ X ν = ν b − X k = a A k ( t ) t + s + k (cid:12)(cid:12)(cid:12)(cid:12) s = ν = − b − X k = a A k ( t ) k − X l = a t + l + ν has only poles possible at t = a ∗ − b + 1 , a ∗ − b + 2 , . . . , b ∗ − a − . Since H ( t )( t + l + ν ) (cid:12)(cid:12) t = − l − ν = b X k = l +1 A k ( − l − ν ) = I ( t )( t + l + ν ) (cid:12)(cid:12) t = − l − ν when 1 − a ∗ ν − b ∗ , it follows that the set of double poles of H ( t ) coincideswith the set of simple poles of I ( t ), and therefore is also contained at integers in[ a ∗ − b + 1 , b ∗ − a − H ( t ) may still possess simple poles at integers in[ a ∗ − b + 1 , b ∗ − a − e H ( t ) = e H ( a , b ; t ) = ( t + b ) a − b ( a − b )! ( t + b ) a − b ( a − b )! ( t + b ) a − b ( a − b )! H ( t )= b − a ∗ − X j =1+ a − b ∗ B j ( t + j ) + b − a ∗ − X j =1+ a − b ∗ C j t + j , (15)because the rational function e H ( t ) has degree at most − t + b ) a − b ( a − b )! ( t + b ) a − b ( a − b )! ( t + b ) a − b ( a − b )!has at least simple zeroes at t = 1 − a ∗ , . . . , − b ∗ and at least double zeroes at t = 1 − a ∗ , − a ∗ , . . . , − b ∗ and taking into account condition (10), we find out that e H ( t ) does not have poles in the half-plane Re t > c , hence the expansion (15)‘shortens’ to e H ( t ) = b − a ∗ − X j = a ∗ B j ( t + j ) + b − a ∗ − X j = a ∗ C j t + j . In fact, the second sum is over the interval max { a ∗ , a − b ∗ } j b − a ∗ − { a ∗ , a − b ∗ } j b − a ∗ − ν = 1 − a ∗ ) in mind, we conclude that the coefficients B j = e H ( t )( t + j ) and C j = ∂∂t (cid:0) e H ( t )( t + j ) (cid:1)(cid:12)(cid:12)(cid:12) t = − j for j ∈ Z satisfy B j ∈ Z , d m C j ∈ Z with m = max { a − b , a − b , a − b , a − b , a − b , a − b } , YPERGEOMETRIC RATIONAL APPROXIMATIONS TO ζ (4) 13 but alsoord p B j , ord p ( d m C j ) ≥ min j,k ∈ Z (cid:18)(cid:22) b − a − p (cid:23) − (cid:22) k − a p (cid:23) − (cid:22) b − k − p (cid:23) + X r ∈{ , , } (cid:18)(cid:22) j − b r p (cid:23) − (cid:22) j − a r p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19) + X r ∈{ , , } (cid:18)(cid:22) k − j − b r p (cid:23) − (cid:22) k − j − a r p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19)(cid:19) for primes p > √ b − a (see [20, Lemmas 17, 18]). Furthermore,12 πi Z c + i ∞ c − i ∞ e H ( t ) (cid:18) π sin πt (cid:19) cos πt d t = ∞ X ν =1 − a ∗ (cid:18) b − a ∗ − X j = a ∗ B j ( ν + j ) + b − a ∗ − X j = a ∗ C j ( ν + j ) (cid:19) = 3 ζ (4) b − a ∗ − X j =max { a ∗ , a − b ∗ } B j − (cid:18) b − a ∗ − X j = a ∗ B j j − a ∗ X l =1 l + b − a ∗ − X j = a ∗ C j j − a ∗ X l =1 l (cid:19) , where P j C j = 0 was implemented. Performing the same way for the second doubleintegral in (11) we conclude that G ( a , b ) = B ( a , b ) ζ (4) − C ( a , b ) , where B ∈ Z , d m d m C ∈ Z (16)with m = max { b − a ∗ − a ∗ − , b − a ∗ − a ∗ − } ,m = max { b − a ∗ − a ∗ − , b − a ∗ − a ∗ − , a − b , . . . , a − b } , andord p B, p C ≥ min j,l ∈ Z (cid:18)(cid:22) b − a − p (cid:23) − (cid:22) j + l − a p (cid:23) − (cid:22) b − j − l − p (cid:23) + X r ∈{ , , } (cid:18)(cid:22) j − b r p (cid:23) − (cid:22) j − a r p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19) + X r ∈{ , , } (cid:18)(cid:22) l − b r p (cid:23) − (cid:22) l − a r p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19)(cid:19) (17)for primes p > √ b − a − b ∗ + b ∗ = a −
1) but can be potentiallydropped without significant arithmetic losses. For example, if b ∗ + b ∗ > a − e H ( t ) = b ∗ − X j =1+ a − b ∗ B j ( t + j ) + b − a ∗ − X j = a ∗ B j ( t + j ) + b ∗ − X j =1+ a − b ∗ C j t + j + b − a ∗ − X j = a ∗ C j t + j , so that there are poles of e H ( t ) to the right of the contour Re t = c . The corre-sponding residues of the integrand areRes t = − j e H ( t ) (cid:18) π sin πt (cid:19) cos πt = D j − ζ (4) B j with D j = 124 ∂ ∂t (cid:0) e H ( t )( t + j ) (cid:1)(cid:12)(cid:12)(cid:12) t = − j = 12 ∂ ∂t (cid:18) e H ( t ) − B j ( t + j ) − C j t + j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t = − j , where j is an integer in the interval 1 + a − b ∗ j b ∗ − (cid:18) π sin πt (cid:19) cos πt = 1( t + j ) − ζ (4) ( t + j ) + O (cid:0) ( t + j ) (cid:1) as t → − j. Proceeding as above we deduce that12 πi Z c + i ∞ c − i ∞ e H ( t ) (cid:18) π sin πt (cid:19) cos πt d t = ∞ X ν =1 − a ∗ Res t = ν e H ( t ) (cid:18) π sin πt (cid:19) cos πt = − ζ (4) b ∗ − X j =1+ a − b ∗ B j + 3 b ∗ − X j =1+ a − b ∗ B j ∞ X l = j +1 − a ∗ l =0 l + 3 b − a ∗ − X j = a ∗ B j ∞ X l = j +1 − a ∗ l + b ∗ − X j =1+ a − b ∗ C j ∞ X l = j +1 − a ∗ l =0 l + b − a ∗ − X j = a ∗ C j ∞ X l = j +1 − a ∗ l , which is again seen to be a linear form in Z ζ (4) + Q .6. The group structure for ζ (4)Following [19], to any set of parameters h from Section 4 we assign the 27-elementmultiset of nonnegative integers e j = h j − , e j = h j + h − − h − j ,e jk = h − h j − h k for 1 j < k , (18)and set H ( e ) = F ( h ) for the quantity defined in that section. By the construction, γ ( h ) − F ( h ) = e ! e ! e ! e ! e ! e ! e ! e ! H ( e )is invariant under any permutation of the parameters h , h , . . . , h (which we canview as the ‘ h -trivial’ action). Clearly, any such permutation induces the corre-sponding permutation of the parameter set (18).On the other hand, it is seen from (6) that the quantity Y j =1 ( a j − b j )! · G ( a , b )does not change when the parameters in either collection a , a , a or a , a , a per-mute; we can regard such permutations as ‘ a -trivial’. (The same effect is produced YPERGEOMETRIC RATIONAL APPROXIMATIONS TO ζ (4) 15 by ‘ b -trivial’ permutations, when we change the order in b , b , b or b , b , b .) Wecan also add to the list the ‘trivial’ involution i : a j ↔ a − j , b j ↔ b − j for j = 1 , . . . , , which reflects the symmetry s ↔ t of the double integral (6). In addition, werecall that G ( a , b ) is left unchanged by the simultaneous shifts of a , a , a , a and b , b , b , b (or of a , a , a , a and b , b , b , b , respectively) by the same integer. Weregard the action of all these transformations (permutations, shifts and involution)and their compositions as the ‘( a , b )-trivial’ action.By setting a = 1 + h − h − , a j = h j for j = 1 , . . . , ,b = 1 + h , b = b = 1 , b = b = 1 + h − h − ,b = h + h + h − h , b = h + h + h − h . (19)we have F ( h ) = G ( a , b ). If we request the condition h − = 2 + 3 h − ( h + h + h + h + h + h ) (20)to hold, then the shift of h , h , h , h by 1 + h − h − h − h , that is, the trans-formation b : h (1 + 2 h − h − h − h , h − ; 1 + h − h − h , h , h − h − h , h , h − h − h , h ) , induces the composition of the shift of a , a , a , a and b , b , b , b by 1 + h − h − h − h and the permutation ( b b )( b b ). Therefore b , which also induces thepermutation b = ( e e )( e e )( e e )( e e )( e e )( e e )on the parameter set (18), is an ( a , b )-trivial transformation. As a consequence, thequantity Y j =1 ( a j − b j )! · G ( a , b ) = e ! e ! e ! e ! e ! e ! H ( e )does not change by the action of the permutation b . We remark that (20) is a verynatural condition for the application of the ( a , b )-trivial action to F ( h ). Indeed, by(19) we have b − b = b − b , and (20) is equivalent to b − b = b − b , or to b − b = b − b .Taking the multiset E = { e , e , e , e , e , e , e , e , e , e , e , e } we conclude that the quantity H ( e ) Q e ∈E e ! = 1 Q j =1 e j ! e j ! · e ! e ! e ! e ! e ! e ! e ! e ! H ( e ) is invariant under the h -trivial permutations and, by H ( e ) Q e ∈E e ! = Q j =1 ( a j − b j )! · G ( a , b ) e ! e ! e ! e ! e ! e ! Q j =1 e j ! e j ! , also under the permutation b .The permutation group of the multiset (18), which is generated by all h -trivialand 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 meanswe also recover the invariance of the quantity H ( e )Π( e ) , where Π( e ) = e ! e ! e ! e ! e ! e ! e ! e ! e ! e ! e ! e ! , under the action of G and corresponding to the arithmetic normalisation of H ( e ) = F ( h ) = G ( a , b ) in Section 4.Because our access to the arithmetic of coefficients of linear forms H ( e ) ∈ Z ζ (4) + Q is performed through their G ( a , b )-representation, we will be interested in collect-ing a set of representatives which are distinct modulo ( a , b )-trivial transformations.For a generic set of integral parameters h subject to (20), such set of representativescontains 120 different elements. Indeed, by (19) and (20) the subgroup of all the( a , b )-trivial permutations in G contains 3! ·
2! = 432 elements, and is generatedby: • the a - and h -trivial permutations ( h h ) and ( h h ); • the a - and h -trivial permutations ( h h ) and ( h h ); • the b -trivial permutation ( b b )( b b ) (that is, by b ); and • the involution i (that is, by ( h h )( h h )( h h )).This subgroup also contains ( b b )( b b ) (namely, b = ib i ), and is isomorphicto S × S . Now, the group G is generated by ( h h ), ( h h ), ( h h ), ( h h ), b and ( h h ). Note that ( h h ), ( h h ), ( h h ) and ( h h ) commute with b ,while ( h h ) acts on ( a , b ) by a ↔ a , b b + a − a , b b + a − a (andleaves a i , b i unchanged for i = 3 , | G | /
432 = 120 elementsin G that are distinct modulo the ( a , b )-trivial subgroup, each for any simultaneouschoice of a subset { a , a , a } (or { a , a , a } ) of { h , . . . , h } (among all (cid:0) (cid:1) = 20such subsets) and of a permutation in the b -trivial subgroup (of 3! = 6 elements)generated by b and b .7. Arithmetic of linear forms
In order to compute the minimum on the right-hand side of (17), we distinguishtwo different situations: (a) j + l − a is coprime with p , and (b) j + l − a is divisibleby p . In case (a), we get ⌊ ( j + l − a ) /p ⌋ = ⌊ ( j + l − a − /p ⌋ , so that the minimum YPERGEOMETRIC RATIONAL APPROXIMATIONS TO ζ (4) 17 in (17) is greater or equal thanΩ ( a , b ; p ) = min j,l ∈ Z (cid:18)(cid:22) b − a − p (cid:23) − (cid:22) j + l − a − p (cid:23) − (cid:22) b − j − l − p (cid:23) + X r ∈{ , , } (cid:18)(cid:22) j − b r p (cid:23) − (cid:22) j − a r p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19) + X r ∈{ , , } (cid:18)(cid:22) l − b r p (cid:23) − (cid:22) l − a r p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19)(cid:19) . (21)In case (b), we have l = − j + a + µp for some µ ∈ Z and (cid:22) b − a − p (cid:23) − (cid:22) j + l − a p (cid:23) − (cid:22) b − j − l − p (cid:23) + X r ∈{ , , } (cid:18)(cid:22) j − b r p (cid:23) − (cid:22) j − a r p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19) + X r ∈{ , , } (cid:18)(cid:22) l − b r p (cid:23) − (cid:22) l − a r p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19) = X r ∈{ , , } (cid:18)(cid:22) j − b r p (cid:23) − (cid:22) j − a r p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19) + X r ∈{ , , } (cid:18)(cid:22) − j + a − b r p (cid:23) − (cid:22) − j + a − a r p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19) = X r ∈{ , , } (cid:18)(cid:22) j − b r p (cid:23) − (cid:22) j − a r p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19) + X r ∈{ , , } (cid:18)(cid:22) j + a r − a − p (cid:23) − (cid:22) j + b r − a − p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19) for primes p > √ b − a −
2, where the property ⌊ α + µ ⌋ = ⌊ α ⌋ + µ was used,together with the passage (cid:26) a − p (cid:27) + (cid:26) − ap (cid:27) = p − p , where a ∈ Z , for the fractional part { α } = α − ⌊ α ⌋ of a number. This means that in case (b) theminimum in (17) is equal toΩ ( a , b ; p ) = min j ∈ Z (cid:18) X r ∈{ , , } (cid:18)(cid:22) j − b r p (cid:23) − (cid:22) j − a r p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19) + X r ∈{ , , } (cid:18)(cid:22) j + a r − a − p (cid:23) − (cid:22) j + b r − a − p (cid:23) − (cid:22) a r − b r p (cid:23)(cid:19) . (22) Combining the two cases together we conclude thatord p B ( a , b ) , ord p d m d m C ( a , b ) ≥ min { Ω ( a , b ; p ) , Ω ( a , b ; p ) } for primes p > √ b − a −
2, where the quantities Ω and Ω are defined in (21)and (22).When we choose a j = α j n + 1 for j = 0 , , . . . , ,b = β n + 3 and b j = β j n + 1 for j = 1 , . . . , , (23)for some positive set of integer directions ( α , β ), then computing Ω , Ω reduces tothe computation of the minima ω ∗ ( x ) and ω ∗ ( x ) of functions ω ( x, y, z ) = ⌊ ( β − α ) x ⌋ − ⌊ y + z − α x ⌋ − ⌊ β x − ( y + z ) ⌋ + X r ∈{ , , } (cid:0) ⌊ y − β r x ⌋ − ⌊ y − α r x ⌋ − ⌊ ( α r − β r ) x ⌋ (cid:1) + X r ∈{ , , } (cid:0) ⌊ z − β r x ⌋ − ⌊ z − α r x ⌋ − ⌊ ( α r − β r ) x ⌋ (cid:1) and ω ( x, y ) = X r ∈{ , , } (cid:0) ⌊ y − β r x ⌋ − ⌊ y − α r x ⌋ − ⌊ ( α r − β r ) x ⌋ (cid:1) + X r ∈{ , , } (cid:0) ⌊ y + ( α r − α ) x ⌋ − ⌊ y + ( β r − α ) x ⌋ − ⌊ ( α r − β r ) x ⌋ (cid:1) over y, z and over y , respectively. Indeed,Ω ( a , b ; p ) = ω (cid:18) np , j − p , l − p (cid:19) and Ω ( a , b ; p ) = ω (cid:18) np , j − p (cid:19) in the settings above. This means thatord p B ( a , b ) , p C ( a , b ) ≥ min (cid:26) ω ∗ (cid:18) np (cid:19) , ω ∗ (cid:18) np (cid:19)(cid:27) (24)for primes p > p ( β − α ) n .Notice that the functions ω and ω (hence their minima) are 1-periodic in eachvariable, so it is sufficient to compute them on the intervals [0 , α = β = · · · = β = 0 , α = · · · = α = 1 and β = 3we already get (by droping the four non-negative terms in both ω and ω ) ⌊ x ⌋ − ⌊ y + z ⌋ − ⌊ x − ( y + z ) ⌋ + ( ⌊ y ⌋ − ⌊ y − x ⌋ − ⌊ x ⌋ ) + ( ⌊ z ⌋ − ⌊ z − x ⌋ − ⌊ x ⌋ )= ⌊ x ⌋ − ⌊ y + z ⌋ − ⌊ x − ( y + z ) ⌋ − ⌊ y − x ⌋ − ⌊ z − x ⌋ ≥ x ∈ [ , YPERGEOMETRIC RATIONAL APPROXIMATIONS TO ζ (4) 19 and ( ⌊ y ⌋ − ⌊ y − x ⌋ − ⌊ x ⌋ ) + ( ⌊ y + x ⌋ − ⌊ y ⌋ − ⌊ x ⌋ )= ⌊ y + x ⌋ − ⌊ y − x ⌋ ≥ x ∈ [ , . Indeed, when x ∈ [ , ⌊ x ⌋ − ⌊ y + z ⌋ − ⌊ x − ( y + z ) ⌋ ≥ ⌊ y − x ⌋ + ⌊ z − x ⌋ − y < x or z < x ; otherwise, x y < x z < ⌊ x ⌋ = 2 , ⌊ y + z ⌋ = 1 , ⌊ x − ( y + z ) ⌋ = ⌊ x − ( y − x ) − ( z − x ) ⌋ = 0and ⌊ y − x ⌋ = ⌊ z − x ⌋ = 0 . A proof of the second inequality, when x ∈ [ , ⌊ y − x ⌋ = − ⌊ y + x ⌋ ≥ y < , and of ⌊ y + x ⌋ = 1, ⌊ y − x ⌋ y <
1. The twoinequalities together mean that the quantity Φ n from Section 3 is divisible by (4).It looks quite plausible (though we do not possess any proof of this) that we alwayshave ω ∗ ( x ) ≥ ω ∗ ( x ) except for possibly finitely many rational points on the interval[0 , ω ( x, y, α x − y ) coincides with ω ( x, y ) apart from finitely manyrational lines crossing the square [0 , .)Now assume that the linear forms G ( a , b ) originate from the forms F ( h ) of Sec-tion 4 and condition (20) written as2 β + α = α + α + α + α + α + α holds. In this case we can scale all the parameters in (18) to discuss the set e n instead, where e j = α j , e j = α j − α for 1 j ,e jk = β − α j − α k for 1 j < k , and record the related quantities by H ( e n ) = B ( e n ) ζ (4) − C ( e n ) ∈ Z ζ (4) + Q ,where n = 0 , , , . . . . The discussion above (see (24)) implies thatord p B ( e n ) , p C ( e n ) ≥ ω ∗ (cid:18) e ; np (cid:19) for primes p > p ( β − α ) n, where ω ∗ ( e ; x ) = min { ω ∗ ( e ; x ) , ω ∗ ( e ; x ) } , hence alsoord p B ( g e n ) , p C ( g e n ) ≥ ω ∗ (cid:18) g e ; np (cid:19) for primes p > p ( β − α ) n and g ∈ G , where g e denotes the image of the multiset e under the action of g ∈ G . At thesame time, B ( e n )Π( e n ) = B ( g e n )Π( g e n ) and C ( e n )Π( e n ) = C ( g e n )Π( g e n ) for all g ∈ G , in view of the invariance of H ( e n ) / Π( e n ) under the action of G (andof the irrationality of ζ (4)). This implies thatord p B ( e n ) , p C ( e n ) ≥ ord p (cid:18) Π( e n )Π( g e n ) ω ∗ (cid:18) g e ; np (cid:19)(cid:19) = X e ∈E (cid:18)(cid:22) enp (cid:23) − (cid:22) g enp (cid:23)(cid:19) + ω ∗ (cid:18) g e ; np (cid:19) for primes p > p ( β − α ) n and all g ∈ G , henceord p B ( e n ) , p C ( e n ) ≥ ω (cid:18) e ; np (cid:19) for primes p > p ( β − α ) n, where ω ( e ; x ) = max g ∈ G (cid:18)X e ∈E ( ⌊ ex ⌋ − ⌊ g ex ⌋ ) + ω ∗ ( g e ; x ) (cid:19) . (25)The maximum can be restricted to distinct representatives modulo the group of( a , b )-trivial permutations.8. One concrete example of irrationality measure for ζ (4)In the notation of Section 4 we take h = η n + 2 , h − = η − n + 2 , h = η n + 1 , . . . , h = η n + 1with η = ( η , η − ; η , . . . , η ) = (68 ,
57; 22 , , , , , . If we set F n = F ( h ) = G ( a , b ) = u n ζ (4) − v n then the asymptotics of F n and u n as n → ∞ is computed with the help of [19, Proposition 1] (adapted here to address aa slightly different normalisation of F ( h )): C = − lim n →∞ log | F n | n = 36 . . . . and C = lim n →∞ log | u n | n = 106 . . . . . The above choice of h translates the form F n = F ( h ) from (5) into G ( a , b ) from(6) with the parameters (23) as follows: α = (11; 22 , , , , , , β = (68; 0 , , , , , . (26)The denominator of v n = C ( a , b ) in (16) is d n d n . The following table lists 31out of 120 representatives under the action of group G on (26) modulo the trivial( a , b )-action, only those that contribute to the computation of the correspondingfunction ω ( x ) = ω ( e ; x ) in (25): YPERGEOMETRIC RATIONAL APPROXIMATIONS TO ζ (4) 211 (68; 22, 23, 24, 25, 26, 27)2 (68; 22, 23, 24, 25, 27, 26)3 (68; 22, 23, 24, 26, 25, 27)4 (68; 22, 23, 25, 24, 26, 27)5 (68; 22, 23, 25, 24, 27, 26)6 (68; 22, 23, 26, 24, 27, 25)7 (68; 22, 24, 23, 25, 26, 27)8 (68; 22, 24, 23, 25, 27, 26)9 (68; 22, 25, 23, 26, 24, 27)10 (67; 21, 22, 23, 25, 26, 27) 11 (67; 21, 22, 23, 25, 27, 26)12 (67; 21, 22, 23, 26, 25, 27)13 (67; 21, 22, 25, 23, 26, 27)14 (66; 20, 21, 23, 24, 26, 27)15 (66; 20, 21, 23, 24, 27, 26)16 (66; 20, 21, 23, 26, 24, 27)17 (66; 20, 21, 24, 23, 27, 26)18 (65; 19, 20, 23, 24, 25, 27)19 (65; 19, 20, 23, 24, 27, 25)20 (65; 19, 20, 23, 25, 24, 27)21 (65; 19, 20, 24, 23, 27, 25) 22 (65; 19, 21, 22, 23, 26, 27)23 (65; 19, 21, 22, 23, 27, 26)24 (65; 19, 21, 23, 22, 26, 27)25 (65; 19, 21, 23, 22, 27, 26)26 (65; 19, 21, 26, 22, 27, 23)27 (65; 19, 22, 21, 23, 26, 27)28 (65; 19, 23, 20, 24, 27, 25)29 (64; 18, 19, 23, 25, 24, 26)30 (64; 19, 20, 21, 22, 27, 26)31 (64; 19, 21, 20, 22, 26, 27) Here we give the representatives in the format ( β ; α , . . . , α ) = ( η ; η , . . . , η ); allother parameters are completely determined by the data.Then ω ( x ) = 0 if x ∈ (cid:2) , (cid:1) ∪ (cid:2) , (cid:1) ∪ (cid:2) , (cid:1) ∪ (cid:2) , (cid:1) ∪ (cid:2) , (cid:1) ∪ (cid:2) , (cid:1) ,ω ( x ) = 1 if x ∈ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ,ω ( x ) = 2 if x ∈ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ,ω ( x ) = 3 if x ∈ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ,ω ( x ) = 4 if x ∈ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ,ω ( x ) = 5 if x ∈ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i ∪ (cid:2) , (cid:1) h i , YPERGEOMETRIC RATIONAL APPROXIMATIONS TO ζ (4) 23 where the notation [ a, b ) h N i means that the maximum in (25) is attained on the N -th representative. Denoting Φ n = Y p> √ n p ω ( n/p ) we conclude that Φ − n u n ∈ Z and Φ − n d n d n v n ∈ Z ; in other words,Φ − n d n d n F n ∈ Z ζ (4) + Z for n = 1 , , . . . . At the same time the asymptotics of Φ − n d n d n is controlled by the prime numbertheorem: C = lim n →∞ log(Φ − n d n d n ) n = 3 ·
21 + 23 − Z ω ( x ) d ψ ( x ) = 25 . . . . , where ψ ( x ) is the logarithmic derivative of the gamma function. Now [19, Propo-sition 3] applies to imply that the irrationality exponent of ζ (4) is bounded aboveby C + C C − C = 12 . . . . . Finally, we point out that the general family of rational approximations to ζ (4)from Section 5 is only exploited here when it is linked to the old approximationsreviewed in Section 4. A reason behind this is mainly an easy access to the asymp-totic behaviour of the corresponding forms G ( a , b ) and their coefficients B ( a , b ).One may hope to get a better control of general approximations from Section 5 bycovering analytic aspects of the 12-parametric family there, however this will notnecessarily lead to (significantly) better arithmetic consequences. References [1]
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C. Viola , The group structure for ζ (3), Acta Arith. :3 (2001), 269–293.[16] T. Rivoal , La fonction zˆeta de Riemann prend une infinit´e de valeurs irrationnelles auxentiers impairs
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J.London Math. Soc. (2) :1 (2004), 215–230.[22] W. Zudilin , Two hypergeometric tales and a new irrationality measure of ζ (2), Ann. Math.Qu´e. :1 (2014), 101–117. Dipartimento di Ingegneria e Geologia, Universit`a di Chieti-Pescara, Viale Pin-daro, 42, 65127 Pescara, Italy
E-mail address : [email protected] Department of Mathematics, IMAPP, Radboud University, PO Box 9010, 6500 GLNijmegen, Netherlands
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