Plethystic logarithm

The plethystic logarithm is an operator which is the inverse of the plethystic exponential.

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

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

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

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

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

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

• Abramowitz, Milton; Stegun, Irene A. (1972) [1964]. Handbook of mathematical functions: with formulas, graphs and mathematical tables [conference under the auspices of the National science foundation and the Massachusetts institute of technology]. Dover books on advanced mathematics. New York: Dover. ISBN 978-0-486-61272-0.