具增长次数的Riemann xi函数的Taylor多项式根的动力学行为
Dynamic behavior of the roots of the Taylor polynomials of the Riemann xi function with growing degree
摘要：
我们建立了Riemann函数xi的Taylor多项式的一致逼近结果，该结果在整个复平面上都是有效的.特别是，我们确定了一个域增长的多项式的程度上，他们收敛到黎曼的xi函数。使用这种近似，我们得到的“假零”的泰勒多项式的临界带外，这导致了黎曼-冯Mangoldt型的公式的临界带内的泰勒多项式的零点的数量的数量的估计。超指数收敛的Hurwitz零的泰勒多项式的有界零的xi函数的建立，最后，我们解释我们的逼近技术可以扩展到一个集合的解析L-函数。
英文摘要：
We establish a uniform approximation result for the Taylor polynomials of the xi function of Riemann which is valid in the entire complex plane as the degree grows. In particular, we identify a domain growing with the degree of the polynomials on which they converge to Riemann's xi function. Using this approximation we obtain an estimate of the number of "spurious zeros" of the Taylor polynomial which are outside of the critical strip, which leads to a Riemann - von Mangoldt type of formula for the number of zeros of the Taylor polynomials within the critical strip. Super-exponential convergence of Hurwitz zeros of the Taylor polynomials to bounded zeros of the xi function are established along the way, and finally we explain how our approximation techniques can be extended to a collection of analytic L-functions.
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