Chebyshev functions

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

$\vartheta (x)={\displaystyle \sum _{p\le x}}\log p.$

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

$\psi (x)={\displaystyle \sum _{p\le x}}k\log p,$

where the same summation notation is used and $k$ denotes the unique integer such that $p^k\le x$ but $p^{k+1}>x$. Heuristically, the first of these two functions 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 owe their proof to a relatively 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 .

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 $\psi (x)\sim x$.