用Pochhammer多项式展开分析黎曼ζ函数零点
Analysis of Riemann Zeta-Function Zeros using Pochhammer Polynomial Expansions
摘要：
Riemann函数Xi（t）属于一类整函数，它可以展开为对称Pochhammer多项式的一致收敛级数，依赖于一个实尺度参数β。可以证明，多项式逼近Xi（n，t，beta）到Xi（t）仅在渐近标度极限beta->无穷大中具有不同的实根。因此，可以推断对于n->无穷大，存在递增的β序列beta（n）->无穷大，使得Xi（n，t，beta（n））仅对所有n具有实根，并且对于每个整函数，可以关联满足特定差分方程的唯一最小β序列。数值分析表明与Riemann Xi（t）相关的最小beta（n）序列具有明显的亚对数增长率，并且可以证明当n->无穷大时，如果beta（n）=o（log（n）），则近似Xi（n，t，beta（n））收敛于Xi（t）.证明了复分析中的Hurwitz定理，并对最小β序列的渐近性质进行了形式化分析，确定了一个基本机制，从而有力地证实了黎曼假设的有效性.
英文摘要：
The Riemann Xi-function Xi(t) belongs to a family of entire functions which can be expanded in a uniformly convergent series of symmetrized Pochhammer polynomials depending on a real scaling parameter beta. It can be shown that the polynomial approximant Xi(n,t,beta) to Xi(t) has distinct real roots only in the asymptotic scaling limit beta->infinity. One may therefore infer the existence of increasing beta-sequences beta(n)->infinity for n->infinity, such that Xi(n,t,beta(n)) has real roots only for all n, and to each entire function it is possible to associate a unique minimal beta-sequence fulfilling a specific difference equation. Numerical analysis indicates that the minimal beta(n) sequence associated with the Riemann Xi(t) has a distinct sub-logarithmic growth rate, and it can be shown that the approximant Xi(n,t,beta(n)) converges to Xi(t) when n->infinity if beta(n)=o(log (n)). Invoking the Hurwitz theorem of complex analysis, and applying a formal analysis of the asymptotic properties of minimal beta-sequences, a fundamental mechanism is identified which provides a compelling confirmation of the validity of the Riemann Hypothesis.
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