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1

We have two well-known asymptotics, $$ \begin{align} \sum_{n \leq N} \frac{1}{n} &\sim \log N \\ \sum_{p \leq N} \frac{1}{p} &\sim \log \log N. \end{align}$$ The difference in magnitude between these is so large that $$ \sum_{n \leq X} \frac{\epsilon(n)}{n} \gg \log N$$ still, and so your sum still diverges. More generally, you can hope for ...


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$G^*(x,\xi)=G(\xi,x)$, so $L^*G^*(x,\xi)=L^*G(\xi,x)$. From the properties of Green's function we know that $L^*G^*(x,\xi)=0$ whenever $x$ is an element of $[a,b]$ not equal to $\xi$. Combining these two statements shows that $L^*G(\xi,x)=0$ whenever $\xi$ is an element of $[a,b]$ not equal to $x$, because the $x$ and the $\xi$ have shifted when shifting ...


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Say $f(s)=\sum\frac{a_n}{n^s}$. Define $g(s)=f(s+\sigma_0)$. Then $g(s)=\sum\frac{b_n}{n^s}$ where $b_n=a_n/n^{\sigma_0}\ge0$.



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