L Notation

Related Concepts

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

Definition

For $t,\gamma \in \mathbf{R}$ with 1 ≤ t ≤ 1, the notation L _x [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(L _x [t, γ]) = L _x [t, γ] where π(y) is the number of primes ≤ y

When used to indicate runtimes and for γ fixed, L _x [t, γ] for t...