First Chebyshev function

Definition

Let $x$ be a positive real number. The first Chebyshev function of $x$, denoted $\vartheta (x)$ or $\theta (x)$, is defined as:

$\vartheta (x)=\sum _{p\leq x}\log p$,

where the sum is only over the prime numbers less than or equal to $x$.

Relation with other counting functions

• Prime-counting function: Denoted $\pi (x)$, this simply counts the number of primes less than or equal to $x$.
• Second Chebyshev function: A similar summation, but this time of the von Mangoldt function over all prime powers less than or equal to $x$.