What about $\lim_{n\to\infty}\frac{\sum_{k=1}^n s_k\mu(k)}{n}$, for the zeros of Dirichlet eta function $s_k=1+\frac{2\pi k}{\log 2}i$ with $k\geq 1$?

Question

Let for integers $k\geq 1$ the corresponding zeros of Dirichlet eta function of the form $$s_k=1+\frac{2\pi k}{\log 2}i,$$ then we can consider the following puzzle, when we multiply previous equation by $\mu(k)$, its corresponding values from Mobius function: $$\sum_{k=1}^n s_k\mu(k)=M(n)+\frac{2\pi i}{\log 2}\sum_{k=1}^n\mu(k)k,$$ where $M(n)$ is the evaluation of Mertens function at $x=n$. Thus $$\frac{\sum_{k=1}^n s_k\mu(k)}{n}=\frac{M(n)}{n}+\frac{2\pi i}{\log 2}\frac{\sum_{k=1}^n\mu(k)k}{n}.$$ Now I know that Prime Number Theorem is equivalent to $M(x)=o(x)$, thus exists $\lim_{n\to\infty}\frac{M(n)}{n}$, and this value is $0$. But

Question. What about $$\lim_{n\to\infty}\frac{\sum_{k=1}^n s_k\mu(k)}{n}?$$

This is, you know how analyse the convergence and/or how compute the third limit involving $\frac{\sum_{k=1}^n\mu(k)k}{n}$.

Thanks in advance.

Answer 1

Here's a possible start.

Since, by summation by parts, $\sum_{k=1}^n f_kg_k =f_1\sum_{k=1}^n g_k+\sum_{j=1}^{n-1}(f_{j+1}-f_j)\sum_{k=j+1}^ng_k $, if $f_k = k$, $\sum_{k=1}^n kg_k =\sum_{k=1}^n g_k+\sum_{j=1}^{n-1}\sum_{k=j+1}^ng_k $ so $\frac1{n}\sum_{k=1}^n kg_k =\frac1{n}\sum_{k=1}^n g_k +\frac1{n}\sum_{j=1}^{n-1}\sum_{k=j+1}^ng_k $.

Setting $g = \mu$, $\frac1{n}\sum_{k=1}^n k\mu(k) =\frac1{n}\sum_{k=1}^n \mu(k) +\frac1{n}\sum_{j=1}^{n-1}\sum_{k=j+1}^n\mu(k) $.

I know this is elementary, but it is all I can think of now.