High pseudomoments of the Riemann zeta function
High pseudomoments of the Riemann zeta function
复制标题
黎曼 zeta 函数的高赝矩
DOI：
10.1016/j.jnt.2018.10.001
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发表时间：
2018
影响因子：
0.7
通讯作者：
Winston Heap
中科院分区：
文献类型：
--
作者：
Ole Fredrik Brevig;Winston Heap
Riemann zeta函数的伪矩（记为Mk（N））定义为临界线上N阶部分和的2k阶积分矩。我们改进了估计Mk（N）<$k（log <$N）k2中常数的上、下界，当k≥ 1时N→∞，从而确定了渐近展开式的前两项.我们还研究了k的一致取值范围，其中改进的估计成立，并且当Mk（N）可以由临界线上的N阶部分和的L∞范数的2k次幂下界.
The pseudomoments of the Riemann zeta function, denoted M k (N), are defined as the 2kth integral moments of the Nth partial sum of ζ (s) on the critical line. We improve the upper and lower bounds for the constants in the estimate M k (N)≍ k (log N) k 2 as N→∞ for fixed k≥ 1, thereby determining the two first terms of the asymptotic expansion. We also investigate uniform ranges of k where this improved estimate holds and when M k (N) may be lower bounded by the 2kth power of the L∞ norm of the Nth partial sum of ζ (s) on the critical line.