格和的零点：2。广义黎曼猜想的几何
Zeros of Lattice Sums: 2. A Geometry for the Generalised Riemann Hypothesis
摘要：
研究了正方形格上基本双和的零点的位置。这个和可以用黎曼zeta函数和狄利克雷beta函数的乘积来表示，因此所有非平凡零点都位于临界线上的断言是广义黎曼假设（GRH）的一个特例。它表明，一个新的必要和充分条件，这种特殊情况下的GRH举行的是，一个特定的一组等模和等幅轮廓的比例麦克唐纳函数双和相交的临界线。进一步证明了这些等值线只能相交于复平面上称为“内岛”的离散区域边界上的临界线。数值研究与此几何条件的描述，它表明，对于第一个一万个零点的zeta函数和β函数超过70%的零位于外的内岛，因此将保证躺在临界线的参数在这里。给出了黎曼猜想成立的一个新的充分条件。
英文摘要：
The location of zeros of the basic double sum over the square lattice is studied. This sum can be represented in terms of the product of the Riemann zeta function and the Dirichlet beta function, so that the assertion that all its non-trivial zeros lie on the critical line is a particular case of the Generalised Riemann Hypothesis (GRH). It is shown that a new necessary and sufficient condition for this special case of the GRH to hold is that a particular set of equimodular and equiargument contours of a ratio of MacDonald function double sums intersect only on the critical line. It is further shown that these contours could only intersect off the critical line on the boundary of discrete regions of the complex plane called "inner islands". Numerical investigations are described related to this geometrical condition, and it is shown that for the first ten thousand zeros of both the zeta function and the beta function over 70\% of zeros lie outside the inner islands, and thus would be guaranteed to lie on the critical line by the arguments presented here. A new sufficient condition for the Riemann Hypothesis to hold is also presented.
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