Chebyshev functions

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


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


where the same summation notation is used and k denotes the unique integer such that pkx but pk+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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the statement that ϑ(x)<xlog4.

A somewhat less innocuous result is that the prime number theoremMathworldPlanetmath (i.e., that π(x)xlogx) is equivalent to the statement that ϑ(x)x, which in turn, is equivalent to the statement that ψ(x)x.


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