## You are here

HomeChebyshev functions

## Primary tabs

# 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 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

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

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.

## Mathematics Subject Classification

11A41*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

## Attached Articles

## Corrections

broken by yark ✓

grammar by mathwizard ✓

Small typo by Andrea Ambrosio ✓

## Comments

## Please merge this article with objectid 4020, "Mangoldt summ...

Please merge this article with objectid 4020, "Mangoldt summatory function"