The sixth moment of the Riemann zeta function and ternary additive divisor sums
The sixth moment of the Riemann zeta function and ternary additive divisor sums
复制标题
黎曼 zeta 函数的六次矩和三元加法除数和
DOI：
10.19086/da.22057
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Nathan Ng
中科院分区：
文献类型：
--
作者：
Nathan Ng
Hardy 和 Littlewood 发起了黎曼 zeta 函数在临界线上的 2k$ 级时刻的研究。 1918 年 Hardy 和 Littlewood 建立了二阶矩的渐近公式，1926 年 Ingham 建立了四阶矩的渐近公式。从那时起，就没有对其他时刻进行渐近评估。在本文中，我们研究临界线上 zeta 函数的六阶矩。我们证明，某个三元加法除数和族的猜想公式隐含着临界线上黎曼 zeta 函数六阶矩的带有节能误差项的渐近公式。这为康利和戈内克的启发式论证提供了严格的证明。此外，这为 Conrey、Keating、Farmer、Rubinstein 和 Snaith 关于黎曼 zeta 函数移动矩的猜想提供了一些证据。此外，这还改进了 Ivic 定理，Ivic 基于三重除数函数相关和的猜想公式的假设获得了 zeta 函数六阶矩的上限。
Hardy and Littlewood initiated the study of the $2k$-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. Since then no other moments have been asymptotically evaluated. In this article we study the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula with power savings error term for the sixth moment of the Riemann zeta function on the critical line. This provides a rigorous proof for a heuristic argument of Conrey and Gonek. Furthermore, this gives some evidence towards a conjecture of Conrey, Keating, Farmer, Rubinstein, and Snaith on shifted moments of the Riemann zeta function. In addition, this improves on a theorem of Ivic, who obtained an upper bound for the the sixth moment of the zeta function, based on the assumption of a conjectural formula for correlation sums of the triple divisor function.