Prime Theta Function

The prime theta function is defined as

$$\theta(n)\equiv \sum_{i=1}^n \ln p_i,$$

where $p_i$ is the $i$th Prime. As shown by Bach and Shallit (1996),

$$\theta(n)\sim n.$$

References

Bach, E. and Shallit, J. Algorithmic Number Theory, Vol. 1: Efficient Algorithms. Cambridge, MA: MIT Press, pp. 206 and 233, 1996.