PlanetMath: Chebyshev functions

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (23)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Chebyshev functions

(Definition)

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

less than or equal to

, and

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

where the same summation notation is used and

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

and the second does the same, but weighting each prime in accordance with their logarithmic relationship to

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 , 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 $\psi(x)\sim x$ .

Bibliography

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.

"Chebyshev functions" is owned by Mathprof. [ full author list (2) | owner history (2) ]

(view preamble | get metadata)