We consider partial sums of a weighted Steinhaus random multiplicative function and view this as a model for the Riemann zeta function. We give a description of the tails and high moments of this object. Using these we determine the likely maximum of $T\log T$ independently sampled copies of our sum and find that this is in agreement with a conjecture of Farmer–Gonek–Hughes on the maximum of the Riemann zeta function. We also consider the question of almost sure bounds. We determine upper bounds on the level of squareroot cancellation and lower bounds which suggest a degree of cancellation much greater than this which we speculate is in accordance with the influence of the Euler product.

中文翻译：

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Riemann zeta 函数模型的随机乘法函数和极值的部分和

我们考虑加权 Steinhaus 随机乘法函数的部分和，并将其视为黎曼 zeta 函数的模型。我们描述了这个物体的尾部和高矩。使用这些我们确定可能的最大值$吨\mathrm{日志}吨$独立采样我们的和的副本，发现这与 Farmer-Gonek-Hughes 关于黎曼 zeta 函数的最大值的猜想一致。我们还考虑了几乎确定界限的问题。我们确定平方根抵消水平的上限和下限，这表明抵消程度比我们推测的大得多，这与欧拉乘积的影响一致。