On the value Distribution of the Zeta‐Function on the Critical Line
On the value Distribution of the Zeta‐Function on the Critical Line
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关于Zeta函数在临界线上的值分布
DOI：
10.1112/blms/15.5.513
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发表时间：
1983
期刊：
影响因子：
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通讯作者：
M. Jutila
中科院分区：
文献类型：
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作者：
M. Jutila
即使在黎曼假设的假设下，确定临界线 Res= 7 上黎曼 zeta 函数的数量级也是一个非常困难的问题。本说明的目的是表明相应的“统计”问题可以从本质上得到解决。我们感兴趣的是 l ((" 2+'0l 的估计，其概率为 1；换句话说，当 T 趋于无穷大时，t 区间 [0, T] 的子集的测量应该是 o {T)，其中估计无效。这样的估计在定理 1 的推论中给出，定理 2 表明结果最好是对数意义上的。符号 <^,^> 隐含的常数，和 O (...) 以及要引入的其他常数将始终是绝对的，除非另有说明。
Determining the order of magnitude of Riemann's zeta-function£(s) on the critical line Res= 7 is a very difficult problem, even on the assumption of the Riemann hypothesis. The object of this note is to show that the corresponding" statistical" problem can be essentially solved. We are interested in estimates of l ((" 2+'0l which hold with probability one; in other words, the measure of the subset of the t-interval [0, T], where the estimate fails to be valid, should be o {T) as T tends to infinity. Such an estimate is given in the corollary of Theorem 1, and Theorem 2 shows that the result is in a logarithmic sense best possible. The constants implied by the symbols<^,^>, and O (...), as well as the other constants to be introduced, will be always absolute unless otherwise indicated.