论文标题
关于Riemann Zeta功能对数的迭代积分的价值分布
On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: denseness
论文作者
论文摘要
我们认为在某些垂直和水平线上的$ \logζ$的迭代积分。在这里，功能$ζ$是Riemann Zeta功能。无论是否在临界线上riemann zeta功能的值在复杂的平面中密集，这是一个众所周知的开放问题。在本文中，我们为水平线上迭代积分的值的密集性提供了结果。通过使用此结果，我们在Riemann假设下获得了$ \ int_ {0}^{t} \logζ（1/2 + it'）dt'$的值的密度。此外，我们表明，对于任何$ m \ geq 2 $，关键行上的$ m $ times迭代积分的值的密度相当于Riemann假设。
We consider iterated integrals of $\logζ(s)$ on certain vertical and horizontal lines. Here, the function $ζ(s)$ is the Riemann zeta-function. It is a well known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of $\int_{0}^{t} \log ζ(1/2 + it')dt'$ under the Riemann Hypothesis. Moreover, we show that, for any $m\geq 2$, the denseness of the values of an $m$-times iterated integral on the critical line is equivalent to the Riemann Hypothesis.
