A new lower bound for (3/2)k - Wadim Zudilin

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2 Wadim Zudilinthe estimate ‖(3/2) k ‖ > 0.57434 k for k ≥ 5 using computations from [6]and [9].By modifying Beukers’ construction, namely, considering Padé approximationsto a tail of the series(2)1∞(1 − z) m+1 = ∑n=0( m + nm)z nand studying the explicit p-adic order of the binomial coefficients involved,we are able to proveTheorem 1. The following estimate is valid:∥ 3 k ∥ ∥∥∥∥(> 0.58032) k = 2 −k·0.78512916... for k ≥ K,where K is a certain effective constant.2. Hypergeometric backgroundThe binomial series on the left-hand side of (2) is a special case of thegeneralized hypergeometric series( ∣ )A0 , A(3) 1 , . . . , A q ∣∣∣ ∞∑ (A 0 ) k (A 1 ) k · · · (A q ) kq+1F q z =zB 1 , . . . , B k ,qk!(B 1 ) k · · · (B q ) kwhere(A) k =Γ(A + k)Γ(A)=k=0{A(A + 1) · · · (A + k − 1) if k ≥ 1,1 if k = 0,denotes the Pochhammer symbol (or shifted factorial). The series in (3)converges in the disc |z| < 1, and if one of the parameters A 0 , A 1 , . . . , A qis a non-positive integer (i.e., the series terminates) the definition of thehypergeometric series is valid for all z ∈ C.In what follows we will often use the q+1 F q -notation. We will requiretwo classical facts from the theory of generalized hypergeometric series:the Pfaff–Saalschütz summation formula()−n, A, B(4) 3F 2 C, 1 + A + B − C − n ∣ 1= (C − A) n(C − B) n(C) n (C − A − B) n(see, e.g., [11], p. 49, (2.3.1.3)) and the Euler–Pochhammer integral for theGauss 2 F 1 -series( ) ∫ A, B(5) 2 F 1 C ∣ z Γ(C) 1=t B−1 (1 − t) C−B−1 (1 − zt) −A dt,Γ(B)Γ(C − B) 0provided Re C > Re B > 0 (see, e.g., [11], p. 20, (1.6.6)). Formula (5) isvalid for |z| < 1 and also for any z ∈ C if A is a non-positive integer.
A new lower bound for ‖(3/2) k ‖ 33. Padé approximations of the shifted binomial seriesFix two positive integers a and b satisfying 2a ≤ b. Formula (2) yields(6)( 32) 3(b+1) ( ) 27 b+1 (= = 3 b+1 1 − 1 ) −(b+1)8 9∑∞ ( )( ) b + k 1 k ∑∞= 3 b+1 = 3 b−2a+1b 9k=0k=0∑∞ ( ) b + k= an integer + 3 b−2a+1 3 2(a−k)k=a∑∞ ( a + b + ν≡ 3 b−2a+1 bν=0b)3 −2ν (mod Z).( b + kb)3 2(a−k)This motivates (cf. [4]) constructing Padé approximations to the function∞∑( ) ( ) a + b + ν a + b ∑ ∞(7) F (z) = F (a, b; z) =z ν =bbν=0and applying them with the choice z = 1/9.ν=0(a + b + 1) ν(a + 1) νz νRemark 1. The connection of F (a, b; z) with Beukers’ auxiliary seriesH(ã,˜b; z) from [4] is as follows:∑a−1( ) b + kF (a, b; z) = z((1 −a − z) −b−1 −)z k = H(−a − b − 1, a; z).bk=0Although Beukers considers H(ã,˜b; z) only for ã,˜b ∈ N, his constructionremains valid for any ã,˜b ∈ C and |z| < 1. However the diagonal Padéapproximations, used in [4] for H(z) and used below for F (z), are different.Taking an arbitrary integer n satisfying n ≤ b, we follow the generalrecipe of [5], [13]. Consider the polynomial(8)( ) ( )a + b + n −n, a + nQ n (x) =2F 1 a + b a + b + 1 ∣ xn∑( )( )a + n − 1 + µ a + b + n=(−x) µ =µ n − µµ=0n∑q µ x µ ∈ Z[x]µ=0
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A new lower bound for (3/2)k - Wadim Zudilin

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