Are there results, or can someone point me to reading, on computations of the following sort:

$\lim_n \sum_{2 \le x\le n} e^{i \cdot \pi\cdot \log \log x }$

It would seem that there is reason to believe this should be something like $-\log(n)$. For example, see the graphic on this question which is the visual representation of $\sum_x (-1)^{\omega(x)}$ which for large $x$ is like $\sum_x (-1)^{\log \log (x)} = \sum_x e^{i \cdot \pi\cdot \log \log x }$. From the picture expectation, this looks something like $- \log n$.

To start, we note that we can reduce

$\sum_{2 \le x\le n} e^{i \cdot \pi\cdot \log \log x } = \sum_{2 \le x\le n} \log(x)^{i \cdot \pi} $.

Now this looks like it's related to a "more general zeta function" $\zeta (\log (n), -i\pi)$ (As opposed to the "standard" $\zeta(n,-i\pi)$.

From here, it would seem that the problem is likely out of reach from basically a college undergrad math student. Any textbooks/links for approaching this (hopefully that can ease me into it) would be nice, thanks.

Are there results, or can someone point me to reading, on computations of the following sort:

$\lim_n \sum_{2 \le x\le n} (-1)^ { \log \log x }$

It would seem that there is reason to believe this should be something like $-\log(n)$. For example, see the graphic on this question which is the visual representation of $\sum_x (-1)^{\omega(x)}$ which for large $x$ is like $\sum_x (-1)^{\log \log (x)} = \sum_x e^{i \cdot \pi\cdot \log \log x }$. From the picture expectation, this looks something like $- \log n$.

To start, we note that we can reduce

$\sum_{2 \le x\le n} e^{i \cdot \pi\cdot \log \log x } = \sum_{2 \le x\le n} \log(x)^{i \cdot \pi} $.

Now this looks like it's related to a "more general zeta function" $\zeta (\log (n), -i\pi)$ (As opposed to the "standard" $\zeta(n,-i\pi)$.

From here, it would seem that the problem is likely out of reach from basically a college undergrad math student. Any textbooks/links for approaching this (hopefully that can ease me into it) would be nice, thanks.

Are there results, or can someone point me to reading, on computations of the following sort:

$\lim_n \sum_{2 \le x\le n} e^{i \cdot \pi\cdot \log \log x }$

It would seem that there is reason to believe this should be something like $-\log(n)$. For example, see the graphic on this question which is the visual representation of $\sum_x (-1)^{\omega(x)}$ which for large $x$ is like $\sum_x (-1)^{\log \log (x)} = \sum_x e^{i \cdot \pi\cdot \log \log x }$. From the picture expectation, this looks something like $- \log n$.

I dont know where to start, sources would be nice, thank you!

Are there results, or can someone point me to reading, on computations of the following sort:

$\lim_n \sum_{2 \le x\le n} e^{i \cdot \pi\cdot \log \log x }$

It would seem that there is reason to believe this should be something like $-\log(n)$. For example, see the graphic on this question which is the visual representation of $\sum_x (-1)^{\omega(x)}$ which for large $x$ is like $\sum_x (-1)^{\log \log (x)} = \sum_x e^{i \cdot \pi\cdot \log \log x }$. From the picture expectation, this looks something like $- \log n$.

To start, we note that we can reduce

$\sum_{2 \le x\le n} e^{i \cdot \pi\cdot \log \log x } = \sum_{2 \le x\le n} \log(x)^{i \cdot \pi} $.

Now this looks like it's related to a "more general zeta function" $\zeta (\log (n), -i\pi)$ (As opposed to the "standard" $\zeta(n,-i\pi)$.

From here, it would seem that the problem is likely out of reach from basically a college undergrad math student. Any textbooks/links for approaching this (hopefully that can ease me into it) would be nice, thanks.

Are there results, or can someone point me to reading, on computations of the following:

$\lim_n \sum_{2 \le x\le n} e^{i \cdot \pi\cdot \log \log x }$

It would seem that there is reason to believe this should be something like $-\log(n)$. For example, see the graphic on this question which is the visual representation of $\sum_x (-1)^{\omega(x)}$ which for large $x$ is like $\sum_x (-1)^{\log \log (x)} = \sum_x e^{i \cdot \pi\cdot \log \log x }$. From the picture expectation, this looks something like $- \log n$.

I dont know where to start, sources would be nice, thank you!

Are there results, or can someone point me to reading, on computations of the following sort:

$\lim_n \sum_{2 \le x\le n} e^{i \cdot \pi\cdot \log \log x }$

It would seem that there is reason to believe this should be something like $-\log(n)$. For example, see the graphic on this question which is the visual representation of $\sum_x (-1)^{\omega(x)}$ which for large $x$ is like $\sum_x (-1)^{\log \log (x)} = \sum_x e^{i \cdot \pi\cdot \log \log x }$. From the picture expectation, this looks something like $- \log n$.

I dont know where to start, sources would be nice, thank you!