This paper discusses some infinite trigonometric products which arecharacteristic functions of simple random walks on the real line; in fact,these define "random Riemann-$\zeta$ functions," a notion which is explained.The concept of typicality for random Riemann $\zeta$ functions is explained andconnected to the Riemann hypothesis. Then it is shown that the distributionfunctions of these random walks are Schwarz functions. It is also shown thattheir characteristic function factors into the characteristic function of aLevy stable random variable and a subdominant fluctuating factor. As acorollary it also follows that amateur mathematician Benoit Cloitre's infinitetrigonometric product $\prod_{n=1}^\infty\left[\frac23+\frac13\cos\left(\frac{x}{n^{2}}\right)\right] = e^{- C\,\sqrt{|x|} +\varepsilon(|x|)},$ with $|\varepsilon(|x|)| \leq K |x|^{1/3}$for some $K>0$, and with $ C= \int\frac{\sin\xi^2}{2+\cos\xi^2}{\rm{d}}\xi;$numerically, $C = 0.319905585... \sqrt{\pi}$. This confirms a surmise of BenoitCloitre. The $O\big(|x|^{1/3}\big)$ error bound is empirically found to beaccurate for moderately sized $|x|$ but not for larger $|x|$. This difference$\varepsilon(|x|)$ between Cloitre's $\log \prod_{n\geq1}\left[\frac23+\frac13\cos\left(\frac{x}{n^{2}}\right)\right]$ and its regular trend$-C\sqrt{|x|}$, although deterministic, appears to be an "empiricallyunpredictable" function. Our probabilistic investigation of this phenomenonconnects the fluctuations to the "random Riemann-$\zeta$ function with argument2."
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机译：本文讨论了一些无限三角函数，它们是实线上简单随机游动的特征函数；实际上，这些定义定义为“随机Riemann-$ \ zeta $函数”。对此进行了解释。解释了随机Riemann $ \ zeta $函数的典型性概念并将其与Riemann假设联系起来。然后表明，这些随机游走的分布函数是Schwarz函数。还表明，它们的特征函数因素是Levy稳定随机变量的特征函数和主要的波动因子。当然，业余数学家Benoit Cloitre的无穷三角乘积$ \ prod_ {n = 1} ^ \ infty \ left [\ frac23 + \ frac13 \ cos \ left（\ frac {x} {n ^ {2}} \ right} \ right] = e ^ {-C \，\ sqrt {| x |} + \ varepsilon（| x |）}，$和$ | \ varepsilon（| x |）| \ leq K | x | ^ {1/3} $对于某些$ K> 0 $，并且$ C = \ int \ frac {\ sin \ xi ^ 2} {2+ \ cos \ xi ^ 2} {\ rm {d}} \ xi; $的数值表示$ C = 0.319905585 ... \ sqrt {\ pi} $。这证实了BenoitCloitre的推测。根据经验发现，对于中等大小的$ | x | $，$ O \ big（| x | ^ {1/3} \ big）$错误界限是准确的，但对于较大的$ | x | $而言，错误界限是正确的。 Cloitre的$ \ log \ prod_ {n \ geq1} \ left [\ frac23 + \ frac13 \ cos \ left（\ frac {x} {n ^ {2}} \ right}之间的差异$ \ varepsilon（| x |）$ \ right] $及其规则趋势$ -C \ sqrt {| x |} $尽管具有确定性，但似乎是“凭经验无法预测”的功能。我们对该现象的概率研究将波动与“带有参数2的随机Riemann-$ \ zeta $函数”联系起来。
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