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The question is very easy. Indeed, put $\gamma=\gamma(X,\lambda)$ and $S=\sum_{m \in I} e^{-\gamma x_m}$. The equality $$ \sum_{m \in I} (\lambda-x_m) e^{-\gamma x_m}=0$$ implies $$\lambda S=\sum_{m \ …

I provide my answer to Mathematics.SE cross-post of the question to inform MathOverflow community. Some results from this answer are already in Blue's answer to the cross-post. Following it, we shall …

For each nonnegative integer $n$ we have $$I_n-J_n=\sum_{k=0}^n \frac{a_k}{\ln^{k+1}(\pi)} ( \pi p_k(\ln\pi) - k! )-\sum_{k=0}^n \frac{a_k}{\ln^{k+1}(\pi)} ( S_n p_k(\ln\pi) - k! )=$$ $$(\pi-S_n)\sum_ …

I tried to prove a positive answer by copying and modifying a bit Dap’s answer to your very similar question, so a main contribution to this answer belongs to @Dap. Let $$Z_a=\{x\in[0,a)| \mbox{ the …

Put $N=1$, $M=2$, $\Omega=\Bbb R^N$, and $u(x)=(x,0)$ for each $x\in\Bbb R^N$. Then the graph of $u$ is a straight line, so it has Hausdorff dimension $1=N$. On the other hand, let $C\subset [0,1]$ be …

It seems the following. Dealing with continuous functions on a topological space $X$, it is natural to consider $X$ to be Tychonoff, or, at least, functionally Hausdorff. I recall that a space $X$ i …