Cauchy’s Little Theorem

Cauchy’s Little Theorem for Stieltjes Constants.

Theorem:

\[\begin{equation} \gamma_n = \frac{(-1)^n n!}{2 \pi} \int_{0}^{2 \pi} e^{-nix} \zeta(e^{ix}+1) dx \end{equation}\]

Proof:

Given the Laurent series for the Riemann Zeta function:

\[\begin{equation} \forall z \in \mathbb{C} \setminus \{1\}, \zeta(z) = \frac{1}{z-1} + \sum_{n=1}^\infty \gamma_n \frac{(1-z)^n}{n!} = \frac{1}{z-1} + \sum_{n=1}^\infty \frac{(-1)^n}{n!}(z-1)^n \tag{1} \end{equation}\]

a formal comparison with the Taylor series developed around \(a=1\) yields:

\[\begin{equation} f(x) = \sum_{n=0}^\infty \frac{f^{(n)} (a)}{n!}(x-a)^n \tag{2} \end{equation}\]

which allows us to derive the identity:

\[\begin{equation} (-1)^n \gamma_n = \frac{d^n}{ds^n} \big(\zeta(s)-\frac{1}{s-1}\big) \tag{3} \end{equation}\]

Moreover, if we consider the Cauchy Differentiation Formula:

\[\begin{equation} f^{(n)}(a) = \frac{n!}{2\pi i} \oint_{C} \frac{f(z)}{(z-a)^{n+1}} dz \tag{4} \end{equation}\]

we may deduce that:

\[\begin{equation} (-1)^n \gamma_n = \frac{n!}{2 \pi i} \oint \frac{1}{(s-1)^{n+1}} \big(\zeta(s) - \frac{1}{s-1}\big) ds = \frac{n!}{2 \pi i} \oint \frac{\zeta(s)}{(s-1)^{n+1}} ds \tag{5} \end{equation}\]

Finally, using the change of variables \(s \mapsto 1 + e^{ix}\):

\[\begin{equation} \gamma_n = \frac{(-1)^n n!}{2 \pi} \int_{0}^{2 \pi} e^{-nix} \zeta(e^{ix}+1) dx \end{equation}\]

QED.

References:

1. Hardy, G. H. and Wright, E. M. “The Behavior of zeta(s) when s->1.” §17.3 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 246-247, 1979.

2. Havil, J. Gamma: Exploring Euler’s Constant. Princeton, NJ: Princeton University Press, 2003.

3. Finch, S. R. “Stieltjes Constants.” §2.21 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 166-171, 2003.