Abstract We consider iterated integrals of log ⁡ ζ ⁢ ( s ) {\log\zeta(s)} on certain vertical and horizontal lines. Here, the function ζ ⁢ ( s ) {\zeta(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 ∫ 0 t log ⁡ ζ ⁢ ( 1 2 + i ⁢ t ′ ) ⁢ 𝑑 t ′ {\int_{0}^{t}\log\zeta(\frac{1}{2}+it^{\prime})\,dt^{\prime}} under the Riemann Hypothesis. Moreover, we show that, for any m ≥ 2 {m\geq 2} , the denseness of the values of an m-times iterated integral on the critical line is equivalent to the Riemann Hypothesis.

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关于黎曼 zeta 函数 I 的对数迭代积分的值分布：密度

摘要 我们考虑 log ⁡ ζ ⁢ ( s ) {\log\zeta(s)} 在某些垂直和水平线上的迭代积分。这里，函数 ζ ⁢ ( s ) {\zeta(s)} 是黎曼 zeta 函数。临界线上的黎曼 zeta 函数值在复平面中是否密集是一个众所周知的开放问题。在本文中，我们给出了水平线上迭代积分值的稠密性结果。通过使用这个结果，我们得到 ∫ 0 t log ⁡ ζ ⁢ ( 1 2 + i ⁢ t ′ ) ⁢ 𝑑 t ′ {\int_{0}^{t}\log\zeta(\frac {1}{2}+it^{\prime})\,dt^{\prime}} 在黎曼假设下。此外，我们证明，对于任何 m ≥ 2 {m\geq 2} ，临界线上的 m 次迭代积分的值的密度等价于黎曼假设。