アイテム詳細

要旨

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\nmid n.