椭圆内的原始晶格点
Primitive lattice points inside an ellipse
摘要：
设Q（u，v）是具有任意实系数的正定二元二次型.对于大的实数x，可以求椭圆圆盘Q（u，v）< x中的本原格点（整数点（m，n），gcd（m，n）= 1）的个数B（x），特别是求B（x）的渐近中的余项R（x）。虽然R（x）的上界取决于zeta函数的零自由区域，因此，在大多数已发表的结果中，在黎曼假设，本文处理的较低的估计。证明了R（x）的绝对值在积分意义下至少是一个正常数c乘以x^{1/4}。此外，它显示了如何找到一个明确的值为c，为每个特定的给定形式Q。
英文摘要：
Let Q(u,v) be a positive definite binary quadratic form with arbitrary real coefficients. For large real x, one may ask for the number B(x) of primitive lattice points (integer points (m,n) with gcd(m,n) = 1) in the ellipse disc Q(u,v) < x, in particular, for the remainder term R(x) in the asymptotics for B(x). While upper bounds for R(x) depend on zero-free regions of the zeta-function, and thus, in most published results, on the Riemann Hypothesis, the present paper deals with a lower estimate. It is proved that the absolute value of R(x) is, in integral mean, at least a positive constant c times x^{1/4}. Furthermore, it is shown how to find an explicit value for c, for each specific given form Q.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
