Chebyshev functions
There are two different functions which are collectively known as the Chebyshev functions^{}:
$\vartheta (x)={\displaystyle \sum _{p\le x}}\mathrm{log}p.$ |
where the notation used indicates the summation over all positive primes $p$ less than or equal to $x$, and
$\psi (x)={\displaystyle \sum _{p\le x}}k\mathrm{log}p,$ |
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 $\pi (x)\sim \frac{x}{\mathrm{log}x}$) is equivalent to the statement that $\vartheta (x)\sim x$, which in turn, is equivalent to the statement that $\psi (x)\sim x$.
References
- 1 Ireland, Kenneth and Rosen, Michael. A Classical Introduction to Modern Number Theory^{}. Springer, 1998.
- 2 Nathanson, Melvyn B. Elementary Methods in Number Theory. Springer, 2000.
Title | Chebyshev functions |
---|---|
Canonical name | ChebyshevFunctions |
Date of creation | 2013-03-22 13:50:15 |
Last modified on | 2013-03-22 13:50:15 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 11 |
Author | Mathprof (13753) |
Entry type | Definition |
Classification | msc 11A41 |
Related topic | MangoldtSummatoryFunction |