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 $\vartheta(x)<x\log 4$ .

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 $\psi(x)\sim x$ .

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