On a sequence arising in series for pi (with an appendix by D. Zagier).
Morris Newman, Daniel N. Shanks · MPG.PuRe (Max Planck Society) · 1984
The second author [J. Number Theory 14, 397-423 (1982; Zbl 0492.12004)] introduced a sequence a\\sb n of rational numbers that occur as coefficients in a set of rapidly convergent power series suitable for calculating π. These have the form π =\\frac1\\sqrtN(-\\log \\vert U\\vert -24\\sum\\sp∞\\sbn=1(-1)\\sp n\\fraca\\sb nnU\\sp n) where N is a positive integer and U=U(N) is a real algebraic number determined by N. The remarkably rapid rate of convergence is illustrated by the choice N=3502 in which the first six terms of the series give π correctly to over 500 decimals. The a\\sb n, defined recursively together with a companion sequence of integers c\\sb n, are known to be rational. The present paper develops further properties of the a\\sb n . First, it is shown that they are positive integers and that, in fact, 24 a\\sb n is the coefficient of x\\sp n in the power series expansion of the infinite product \\prod\\sp∞\\sbk=1(1+x\\sp2k-1)\\sp24n . Then it is shown that the a\\sb n satisfy surprising congruences modulo prime powers. For example, a\\sb n is odd if and only if n is a power of 2, and more generally, a\\sbmp\\sp k\\equiv a\\sbmp\\spk-1 (mod p\\sp k) for every prime p and all positive integers m and k. They also derive the inequality (1/3\\sqrtn)(63.87)\\sp nlt;24 a\\sb nlt;64\\sp n, and give a table of the first fifty values of a\\sb n and of c\\sb n. \\par In an appendix, D. Zagier uses the theory of modular forms to strengthen the inequality to an asymptotic series, a\\sb n=C\\frac64\\sp n\\sqrtn(1-\\fracα\\sb 1n+\\fracα\\sb 2n\\sp 2+...)\\quad where\\quad C=\\fracπ12\\fracΓ(3/4)\\sp 2Γ(1/4)\\sp 2. The constants α\\sb 1 and α\\sb 2 are also given explicitly in terms of Γ (3/4) and Γ (1/4). Zagier also uses the modular form description of the a\\sb n to obtain the congruences n a\\sb n\\equiv 1 (mod 3) if 3\ mid n.