On the number of Markoff numbers below a given bound
On the number of Markoff numbers below a given bound
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关于低于给定界限的马尔科夫数的数量
DOI：
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发表时间：
1982
期刊：
影响因子：
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通讯作者：
D. Zagier
中科院分区：
文献类型：
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作者：
D. Zagier
根据马尔科夫的一个著名定理，具有极大极小值（大于判别式平方根的f次方）的不定二次型与迪奥芬图方程p2 + q2 + r1 = ~ipqr的解呈1:1对应关系。通过将求解该方程的Markoffs算法与三角形点阵点计数问题联系起来，证明了小于x的解的数量等于Clog2 3x + 0(log x log log2 x)，其显式可计算常数C = 0.180071704711507 ....给出了101300以内的数值数据，这表明实际误差项要小得多。
According to a famous theorem of Markoff, the indefinite quadratic forms with exceptionally large minima (greater than f of the square root of the discriminant) are in 1 : 1 correspondence with the solutions of the Diophantine equation p2 + q2 + r1 = ~ipqr. By relating Markoffs algorithm for finding solutions of this equation to a problem of count- ing lattice points in triangles, it is shown that the number of solutions less than x equals Clog2 3x + 0(log x log log2 x) with an explicitly computable constant C = 0.18071704711507.... Numerical data up to 101300 is presented which suggests that the true error term is considerably smaller.