L Notation

This is an excerpt from the content

Related Concepts

Exponential Time; O-Notation; Polynomial Time; Subexponential Time

Definition

For \(t,\gamma \in \mathbf{R}\) with 1 ≤ t ≤ 1, the notation Lx[t, γ] is used for any function of x that equals $${e}^{(\gamma +o(1)){(\log x)}^{t}{(\log \log x)}^{1-t} },\mbox{ for }x \rightarrow \infty, $$ where logarithms are natural and where o(1) denotes any function of x that goes to 0 as \(x \rightarrow \infty \) (O notation).

Theory

This function has the following properties:

  • \({L}_{x}[t,\gamma ] + {L}_{x}[t,\delta ] = {L}_{x}[t,\max (\gamma, \delta )]\)

  • \({L}_{x}[t,\gamma ] \cdot {L}_{x}[t,\delta ] = {L}_{x}[t,\gamma + \delta ]\)

  • \({L}_{x}[t,\gamma ] \cdot {L}_{x}[s,\delta ] = {L}_{x}[t,\gamma ]\) if t > s

  • For any fixed k:

  • \({L}_{x}{[t,\gamma ]}^{k} = {L}_{x}[t,k\gamma ]\)

  • If γ > 0 then \({(\log x)}^{k}{L}_{x}[t,\gamma ] = {L}_{x}[t,\gamma ]\)

  • π(Lx[t, γ]) = Lx[t, γ] where π(y) is the number of primes ≤ y

When used to indicate runtimes and for γ fixed, L