Analytic Extension of Polylogarithm Functions and the Riemann Zeta Function

In mathematics, particular emphasis is placed on analytic continuation, which allows us to extend the domain of a function beyond its original definition. In this article, we will explore the analytic continuation of the polylogarithm function, specifically focusing on Li_s(s^{-2} / n^s) and its evaluation at s -1.

Introduction to Polylogarithm Functions

The polylogarithm function, denoted as Li_s(z), is a special function that plays a significant role in various areas of mathematics and physics. It is defined as the infinite series:

Li_s(z) sum_{n1}^{infty} frac{z^n}{n^s}.

This series converges for certain ranges of s and z. An important special case of the polylogarithm function involves the Riemann zeta function, which is Li_s(1) zeta(s).

Functional Equation for Polylogarithm Functions

The polylogarithm function can be extended to the complex plane through a functional equation involving the Hurwitz zeta function. The Hurwitz zeta function, denoted as zeta(s, a), generalizes the Riemann zeta function by including an additional parameter a: zeta(s, a) sum_{n0}^{infty}frac{1}{(n a)^s}.

The functional equation for the polylogarithm function is given by:

Li_s(z) frac{Gamma(1-s)}{2pi^{1-s}} left[i^{1-s} zeta(1-s, frac{1}{2}) frac{1}{2} ln |z| frac{pi}{2} i , text{sgn}(z) right] left[i^{s-1} zeta(1-s, frac{1}{2}) - frac{1}{2} ln |z| frac{pi}{2} i , text{sgn}(z) right]

Evaluating Li_{-1}(1)

Let's consider the specific case where s -1 and z 1 - 1. Substituting these values into the functional equation, we find:

Li_{-1}(1 - 1) frac{Gamma(2)}{2pi^2} left[-zeta(2, frac{1}{2}) - frac{1}{2} ln 0 frac{pi}{2} i , text{sgn}(1 - 1)right] left[-zeta(2, frac{1}{2}) frac{1}{2} ln 0 frac{pi}{2} i , text{sgn}(1 - 1)right]

Note that the logarithm of 0 is undefined in the real number system, which leads to the expression evaluating to infinity. Thus, Li_{-1}(1) infty.

Conclusion

The analytic extension of the polylogarithm function, particularly Li_s(s^{-2} / n^s), provides a powerful tool for exploring complex mathematical structures. While the evaluation of Li_{-1}(1) involves undefined components, it highlights the intricate nature of these functions and the importance of proper domains and limits.

Related Keywords

analytic continuation Polylogarithm function Riemann Zeta function

References

Wikipedia: Polylogarithm

Wikipedia: Hurwitz zeta function