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Let's just deal with the constraint $x_1+\ldots + x_k \le 1$. You can give a pretty good bound as follows: For any $\alpha >0$ the integral you want is at most $$ \frac{1}{k!} \int_{x_1,\ldots,x_k …

One class of functions you could consider is those $f$ for which $|f(x)|+|{\hat f}(x)| \le C(1+|x|)^{-1-\delta}$ for some constant $C$ and some $\delta>0$. Here ${\hat f}(x) = \int_{-\infty}^{\infty} …

If $s=\sigma+it$ with $\sigma >1$ then note that $$ \Big| \frac{\zeta^{\prime}}{\zeta}(s) \Big| =\Big| \sum_{p} \frac{\log p}{p^s-1} \Big| \ge \frac{\log 2}{|2^s-1|} - \sum_{p\ge 3} \frac{\log p}{| …

There are several puzzling things about the question: Firstly of course $\theta$ must be irrational, and it is intended for $\{ x\}$ to denote the Bernoulli polynomial $x-[x]-1/2$ rather than the more …

The left side of your relation is sensitive to whether $\log p_n$ is close to an integer or not, whereas the right side is not. This accounts for the fluctuations. More precisely, the contribution o …