Chebyshev theta function

The Chebyshev $\vartheta$ function is $$\vartheta(x) = \displaystyle\sum_{p \leq x} \log p,$$ where $p \leq x$ denotes that $p$ is a prime number less than the real number $x$.

• Chebyshevthetaplotto50.png

  Plot of $\vartheta$ on $[0,50]$.

• Chebyshevthetaplotto100.png

  Plot of $\vartheta$ on $[0,100]$.

• Chebyshevthetaplotto1000.png

  Plot of $\vartheta$ on $[0,1000]$.