First Chebyshev function

This article is about a function defined on positive reals (and in particular, natural numbers) obtained as the summatory function of an arithmetic function.
View other such summations

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 $x$ .

Modular versions