As I'm waiting for the detailed graphs of \(\nu(s)\) and \(T(-s,2) = P(s)\)... I'll write some graphs on growth.
We recall that:
\[
\begin{align}
q(m) &= (-1)^m\sum_{n \le m} \chi_{m}(n)\\
Q(x) &= \sum_{m \le \lfloor x \rfloor} q(m)\\
\nu(s) &= \sum_{m=1}^\infty q(m) m^{-s}\\
&= -s\int_1^\infty Q(x)x^{-s-1}\,dx\\
\end{align}
\]
Then, we're going to graph \(Q\) in the mean time:
This is a graph of \(Q(x)\) for \(x=1\) up to \(x=1000\). Notably, it never breaches the value \(|Q(x)| > 8\). This function has at most a growth of \(\log(x)\) as I predicted
But it could even be \(O(1)\). The growth \(\log(x)\) or something close, means that \(\nu(s)\) is holomorphic for \(\Re(s) > 0\).
The second function is Gottfried's zeta function, and the growth of its coefficients.
\[
\begin{align}
g(m) &= (-1)^m\sum_{n \le m} \frac{\log(n)^{\frac{\log(m)}{\log(n)}}}{\frac{\log(m)}{\log(n)}!}\chi_{m}(n)\\
A(x) &= \sum_{m \le \lfloor x \rfloor} g(m)\\
P(s) &= \sum_{m=1}^\infty g(m) m^{-s}\\
&= -s\int_1^\infty A(x)x^{-s-1}\,dx\\
\end{align}
\]
The next graph is \(A(x)\) over \(x=1\) upto \(x=1000\), and you will again notice that it is about \(\log(x)\) in growth!
[To be posted when this shit finishes compiling. It is taking wayyyyyyyyy longer than I thought it would. And I don't want to lose this post. Either way, it's going to look like \(Q(x)\), as my calculator is saying. The value \(Q(x)\) is only a little off from \(A(x)\). But the run time to get \(A(x)\) is much longer...]
EDIT: As soon as I wrote that disclaimer, the program compiled 30 mins later:
This is exactly \(\log(x)\). It doesn't break \(|A(x)| > 8\)--it's just a tad more chaotic.
Here is the basic code I am using:
zeta2.gp (Size: 6.4 KB / Downloads: 444)
Open it in a text editor before compiling and running. The read me is in the comments of the source.
Please observe, what I mean is that:
\[
\sup_{0 \le x \le X} |A(x)| \le C\log(X)\\
\]
And, the example I used is:
\[
\sup_{0 \le x \le 1000} |A(x)| \le 8\\
\]
Where at worst, we grow just as slow.........
We recall that:
\[
\begin{align}
q(m) &= (-1)^m\sum_{n \le m} \chi_{m}(n)\\
Q(x) &= \sum_{m \le \lfloor x \rfloor} q(m)\\
\nu(s) &= \sum_{m=1}^\infty q(m) m^{-s}\\
&= -s\int_1^\infty Q(x)x^{-s-1}\,dx\\
\end{align}
\]
Then, we're going to graph \(Q\) in the mean time:
This is a graph of \(Q(x)\) for \(x=1\) up to \(x=1000\). Notably, it never breaches the value \(|Q(x)| > 8\). This function has at most a growth of \(\log(x)\) as I predicted
But it could even be \(O(1)\). The growth \(\log(x)\) or something close, means that \(\nu(s)\) is holomorphic for \(\Re(s) > 0\). The second function is Gottfried's zeta function, and the growth of its coefficients.
\[
\begin{align}
g(m) &= (-1)^m\sum_{n \le m} \frac{\log(n)^{\frac{\log(m)}{\log(n)}}}{\frac{\log(m)}{\log(n)}!}\chi_{m}(n)\\
A(x) &= \sum_{m \le \lfloor x \rfloor} g(m)\\
P(s) &= \sum_{m=1}^\infty g(m) m^{-s}\\
&= -s\int_1^\infty A(x)x^{-s-1}\,dx\\
\end{align}
\]
The next graph is \(A(x)\) over \(x=1\) upto \(x=1000\), and you will again notice that it is about \(\log(x)\) in growth!
[To be posted when this shit finishes compiling. It is taking wayyyyyyyyy longer than I thought it would. And I don't want to lose this post. Either way, it's going to look like \(Q(x)\), as my calculator is saying. The value \(Q(x)\) is only a little off from \(A(x)\). But the run time to get \(A(x)\) is much longer...]
EDIT: As soon as I wrote that disclaimer, the program compiled 30 mins later:
This is exactly \(\log(x)\). It doesn't break \(|A(x)| > 8\)--it's just a tad more chaotic.
Here is the basic code I am using:
zeta2.gp (Size: 6.4 KB / Downloads: 444)
Open it in a text editor before compiling and running. The read me is in the comments of the source.
Please observe, what I mean is that:
\[
\sup_{0 \le x \le X} |A(x)| \le C\log(X)\\
\]
And, the example I used is:
\[
\sup_{0 \le x \le 1000} |A(x)| \le 8\\
\]
Where at worst, we grow just as slow.........
