# Chebyshev functions

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

 $\displaystyle\vartheta(x)=\sum_{p\leq x}\log p.$

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

 $\displaystyle\psi(x)=\sum_{p\leq x}k\log p,$

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 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

 $\displaystyle\prod_{p\leq n}p<4^{n}$

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

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 $\psi(x)\sim x$.

## References

Title Chebyshev functions ChebyshevFunctions 2013-03-22 13:50:15 2013-03-22 13:50:15 Mathprof (13753) Mathprof (13753) 11 Mathprof (13753) Definition msc 11A41 MangoldtSummatoryFunction