plethystic logarithm

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{{notability|date=April 2025}}

{{Short description|Inverse of the plethystic exponential}}

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, $f(t\_1,\; t\_2,\; \backslash cdots,\; t\_n)$, which must equal one at the origin, and is given by{{Cite journal|last1=Benvenuiti |first1=Sergio |last2=Feng |first2=Bo | last3=Hanany | first3=Amihay | last4=He | first4=Yang-Hui |year=2006 |title=Counting BPS Operators in Gauge Theories |page=31 |arxiv= hep-th/0608050v2|quote= |mode= |journal= Journal of High Energy Physics|doi=10.1088/1126-6708/2007/11/050 }}

:$\backslash mathrm\{PL\}[f(t\_1,\; t\_2,\; \backslash cdots,\; t\_n)]\; =\; \backslash sum\_\{k=1\}^\{\backslash infty\}\; \backslash frac\{\backslash mu(k)\}\{k\}\; \backslash text\{ln\}(f(t\_1^k,\; t\_2^k,\; \backslash cdots,\; t\_n^k))$

where $\backslash mu(k)$ is the Möbius function and is defined by {{sfn|Abramowitz|Stegun|1972|p=826}}

:$$\backslash mu(k)\; =$$

\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 $\backslash text\{ln\}(f(t\_1^k,\; t\_2^k,\; \backslash cdots,\; t\_n^k))$ is the natural logarithm of the initial function with every argument raised to the power of $k$.