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On the effective problem of the universality theorem and the denseness problem for zeta and L-functions

遠藤, 健太 名古屋大学

2021.06.17

概要

This thesis is the summary of author’s two studies on the value-distribution of zeta and L-functions.

The first study is on the denseness problem for the iterated integrals of the logarithm of the Riemann zeta-function ζ(s), which is a joint work with Sh¯ota Inoue [9]. We give a result for the denseness of the values of the iterated integrals on horizontal lines. By using this result under the Riemann Hypothesis, we obtain the denseness of the values ∫ t 0 log ζ(1/2 + it′ )dt′ . Moreover, we show that, for any m ≥ 2, the denseness of the values of an m times iterated integral on the critical line is equivalent to the Riemann Hypothesis.

The second study is on the effectivity problem of the universality theorem for zeta and L-functions. Recently, Garunkˇstis, Laurin˘cikas, Matsumoto, J. & R. Steuding showed an effective universality-type theorem for the Riemann zeta-function by using an effective multidimensional denseness result of Voronin. We will generalize Voronin’s effective result and their theorem to the elements of the Selberg class satisfying some conditions.

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