The Stieltjes Constants $\gamma_n$ arise from considering the laurent series of the Riemann Zeta function at $s=1$
$$ \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} (-1)^n \frac{\gamma_n}{n!} (s-1)^n $$
I am curious about the average number of negative signs of the sequence $\gamma_n$ and $b(n)=(-1)^n\gamma_n$.
Are there any known statistics about the sign changes of either of these sequences? I.E. can we possibly say "$\gamma_n$ from $1$ to $n$ is equally negative as it is positive as $n \rightarrow \infty$"?
Precise Question:
We can start by asking if there exists real numbers $0 \le \theta_\gamma^{+}, \theta_{\gamma}^{-1}, \theta_b^{+}, \theta_b^{-1} \le 1$ s.t. we can say
"The proportion $\frac{p}{n}$ of negative numbers in the set $[\gamma_1,... \gamma_n] $ is eventually greater than $\theta_{\gamma}^{-}$ and eventually less than $\theta_{\gamma}^{+}$ as $n \rightarrow \infty$"
Similarly:
"The proportion $\frac{p}{n}$ of negative numbers in the set $[b(1),... b(n)] $ is eventually greater than $\theta_{b}^{-1}$ and eventually less than $\theta_{b}^{+}$ as $n \rightarrow \infty$".
My dream would be a theorem that says
"The proportion $\frac{p}{n}$ of negative numbers in the set $[\gamma_1, ... \gamma_n]$ eventually tends to $\theta_{\gamma}$ as $n \rightarrow \infty$" (and similarly for $B$)
I.E. being able to articulate an exact limiting ratio instead of an upper and lower bound for both $\gamma, b$.
My desire would be if that exact bound would be $\frac{1}{2}$ .
Prior Work:
Krzysztof Maślanka had found a very precise asymptotic bound that appears to closely follow the signs of the Stieltjes Constants (I was made aware of the result from Wikipedia).
In particular they identify that if
$$ \varphi_n =\frac{1}{2}\ln(8\pi) -n+(n + \frac{1}{2})\ln(n) + (s_n - n - \frac{1}{2})\ln(s_n)-\frac{1}{2}\ln(n+s_n)-(c+1)s_n$$
Where $s_n = \frac{n+\frac{3}{2}}{W(\pm \frac{n+\frac{3}{2}}{2\pi i})}$ (I assume the value is invariant under the choice of $\pm$) and $c=\ln(2\pi)+\frac{\pi}{2}i$ then the sign of
$$ \cos(\text{Im}(\varphi_n)) $$
determines the sign of the Stieltjes constants. With some simplification we see we are really interested in understanding:
$$ \cos \left( \text{Im}\left( (s_n - n - \frac{1}{2})\ln(s_n)-\frac{1}{2}\ln(n+s_n)-(c+1)s_n) \right) \right) $$
As I believe the values $s_n$ are complex and $c$ is complex.
While the function experimentally oscillates I am not able to rigorously reason about how often this oscillates from positive to negative. It clearly slows down which is noted in Maślanka's paper. However I don't know "how fast the oscillation is slowing down" and in particular I can't say "the oscillation is slowing down so fast that concrete statements about statistics are not possible".
To expand on this latter point, if the oscillation eventually causes say $r$ negative signs then $2r$ positive signs and then $4r$ negative signs, continuing alternation between powers of 2 (for some natural number $r$) there will simply be no well defined proportion of "negative signs / total" as $n$ tends to infinity (the averages will tend to 0 and 1 endlessly).
