角格和的Riemann假设
The Riemann Hypothesis for Angular Lattice Sums
摘要：
我们提出了进一步的结果，一类涉及复杂的权力的距离点在一个二维正方形晶格和三角函数的角度，补充那些在以前的文件（McPhedran { \em等人}，2008年）的总和。我们给出了一个一般表达式，它允许数值评估的成员的类和任意顺序。我们用它来说明数值的轨迹的实部和虚部的总和为零的属性，我们显示的结果为前两个特定的一组角总和（表示${\cal C}(1,4 m;s)$），这表明他们的密度为零的复指数的临界线是相同的产品（表示${\cal C}(0,1;s)$）的黎曼zeta函数和Catalan β函数。然后我们引入一个函数，它是角格和${\cal C}(1,4 m;s)$与${\cal C}(0,1;s)$的商，并利用它的性质证明了，${\cal C}(1,4 m;s)$服从黎曼假设的任何$m$当且仅当${\cal C}(0,1;s)$服从黎曼假设。我们进一步证明，如果黎曼假设成立，那么${\cal C}(1,4 m;s)$和${\cal C}(0,1;s)$在临界线上具有相同的零点分布（在证明中精确的意义上）。
英文摘要：
We present further results on a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle, supplementing those in a previous paper (McPhedran {\em et al}, 2008). We give a general expression which permits numerical evaluation of members of the class of sums to arbitrary order. We use this to illustrate numerically the properties of trajectories along which the real and imaginary parts of the sums are zero, and we show results for the first two of a particular set of angular sums (denoted ${\cal C}(1,4 m;s)$) which indicate their density of zeros on the critical line of the complex exponent is the same as that for the product (denoted ${\cal C}(0,1;s)$) of the Riemann zeta function and the Catalan beta function. We then introduce a function which is the quotient of the angular lattice sums ${\cal C}(1,4 m;s)$ with ${\cal C}(0,1;s)$, and use its properties to prove that ${\cal C}(1,4 m;s)$ obeys the Riemann hypothesis for any $m$ if and only if ${\cal C}(0,1;s)$ obeys the Riemann hypothesis. We furthermore prove that if the Riemann hypothesis holds, then ${\cal C}(1,4 m;s)$ and ${\cal C}(0,1;s)$ have the same distribution of zeros on the critical line (in a sense made precise in the proof).
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