Polylogarithmic function

Not to be confused with Polylogarithm.

A polylogarithmic function in n is a polynomial in the logarithm of n,

$a_k \log^k(n) + \cdots + a_1 \log(n) + a_0. \,$

In computer science, polylogarithmic functions occur as the order of memory used by some algorithms (e.g., "it has polylogarithmic order").

All polylogarithmic functions are

$P_\ell(x) = o(x^\varepsilon)\,$

for every exponent ε > 0 (for the meaning of this symbol, see small o notation), that is, a polylogarithmic function grows more slowly than any positive exponent. This observation is the basis for the soft O notation Õ(n).