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\begin{document}

\title{Note: Series Reversion of Stieltjes Coefficients}



\author[1]{Benedict Irwin}%
\affil[1]{University of Cambridge}%


\vspace{-1em}



  
  \date{June 18, 2024}


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If we take the Laurent expansion of the Riemann zeta function about $s=1$ $$
\zeta(s) = \frac{1}{s-1} + \sum_{n=0}^\infty \frac{(-1)^n}{n!}\gamma_n (s-1)^n
$$
which defines $\gamma_n$, the Stieltjes constants, where $\gamma_0$ is the Euler-Mascheroni constant. Next perform a series reversion on this to give a series $$
\chi(s) = 1+\frac{1}{s}+\sum_{n=2}^\infty \frac{\kappa(n)}{s^n}
$$
which has expansion \begin{equation}
\chi(s) = 1 + \frac{1}{s} + \frac{\gamma_0}{s^2} + \frac{\gamma_0^2-\gamma_1}{s^3} + \frac{2\gamma_0^3-6\gamma_0\gamma_1+\gamma_2}{2s^4} + \cdots
\label{expansion}
\end{equation}

The coefficients $\kappa(n)$ seem to decrease quite steadily, even up $n$ being a few hundred, where the $\gamma_n$ get large.\selectlanguage{english}
\begin{table}
\begin{tabular}{ |c|c|c| } 
 \hline
 n & \kappa(n) \\
 \hline
0 & 1.000000000\\
1 & 1.000000000\\
2 & 0.5772156649\\
3 & 0.4059937693\\
4 & 0.3135616752\\
5 & 0.2556464523\\
6 & 0.2159181431\\
7 & 0.1869526867\\
8 & 0.1648872027\\
9 & 0.1475121704\\
10 & 0.1334717457\\
11 & 0.1218874671\\
12 & 0.1121649723\\
13 & 0.1038876396\\
14 & 0.09675470803\\
15 & 0.09054358346\\
 \hline
\end{tabular}
\caption{{The first 16 coefficients of the inverse function.}}
\end{table}

Letting $$ 
R_n=\sum_{k=1}^n i_k
$$
and $$
P_n=\sum_{k=1}^n ki_k
$$
and $\{i\}_n=\{i_1,i_2,\cdots|P=n-1\}$, 
I have observed the expression for $\kappa(n)$ from series reversion to be
$$
\kappa(n)=\sum_{\{i\}} (-1)^n\left[\prod_{j=1}^{R-1}j-n\right]\left[\prod_{k=1}^n \frac{1}{i_k!}\left(\frac{\gamma_k}{k!}\right)^{i_{k+1}}\right]\gamma_0^{i_1}
$$
where we define $\kappa(0)=1$. Two examples
$$
\kappa(3) = \gamma_0^2 - \gamma_1 = -\sum_{i_1+2i_2+3i_3=2} \left[\prod_{j=1}^{i_1+i_2+i_3-1} (j-n)\right]\frac{\left(\frac{\gamma_1}{1!}\right)^{i_2}\left(\frac{\gamma_2}{2!}\right)^{i_3}}{i_1!i_2!}\gamma_0^{i_1}
$$
$$
\kappa(4) = \gamma_0^3 - 3 \gamma_0\gamma_1 + \frac{\gamma_2}{2} = \sum_{i_1+2i_2+3i_3+4i_4=3} \left[\prod_{j=1}^{i_1+i_2+i_3+i_4-1} (j-n)\right]\frac{\left(\frac{\gamma_1}{1!}\right)^{i_2}\left(\frac{\gamma_2}{2!}\right)^{i_3}\left(\frac{\gamma_3}{3!}\right)^{i_4}}{i_1!i_2!i_3!}\gamma_0^{i_1}
$$

We can conjecture that $$
\kappa(n+1) < \kappa(n), \;\;\; n\in\mathbb{N}^{>0}?
$$
Is this perhaps a more well behaved way to look at the Stieltjes constants?

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