Draft:Plethystic logarithm

In mathematics, the plethystic logarithm is an operator which is the inverse of the plethystic exponential.

Definition

The plethystic logarithm takes in a function with n complex arguments, $f(t_{1},t_{2},\cdots ,t_{n})$, which must equal one at the origin, and is given by

$\mathrm {PL} [f(t_{1},t_{2},\cdots ,t_{n})]=\sum _{k=1}^{\infty }{\frac {\mu (k)}{k}}{\text{ln}}(f(t_{1}^{k},t_{2}^{k},\cdots ,t_{n}^{k}))$

where $\mu (k)$ is the Möbius function and is defined by

$$\mu (k)={\begin{cases}1&{\text{if }}k=1\\(-1)^{n}&{\text{if }}k{\text{ is the product of }}n{\text{ distinct primes}}\\0&{\text{otherwise}}\end{cases}}$$

and ${\text{ln}}(f(t_{1}^{k},t_{2}^{k},\cdots ,t_{n}^{k}))$ is the natural logarithm of the initial function with every argument raised to the power of $k$.

Applications in Theoretical Physics

The plethystic logarithm has a few applications in theoretical physics, particularly within the study of gauge theories.