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

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.