There are two different functions which are collectively known as the Chebyshev functions:

where the notation used indicates the summation over all positive primes $p$ less than or equal to $x$, and

where the same summation notation is used and $k$ denotes the unique integer such that $p^{k}\leq x$ but $p^{{k+1}}>x$. Heuristically, the first of these two functions measures the number of primes less than $x$ and the second does the same, but weighting each prime in accordance with their logarithmic relationship to $x$.

Many innocuous results in number theory owe their proof to a relatively simple analysis of the asymptotics of one or both of these functions. For example, the fact that for any $n$, we have

is equivalent to the statement that $\vartheta(x)<x\log 4$.

A somewhat less innocuous result is that the prime number theorem (i.e., that $\pi(x)\sim\frac{x}{\log x}$) is equivalent to the statement that $\vartheta(x)\sim x$, which in turn, is equivalent to the statement that $\psi(x)\sim x$.