The Chebyshev function ψ(x), with x < 50

The Chebyshev function ψ(x) − x, for x < 10,000

The Chebyshev function ψ(x) − x, for

The Chebyshev function is either of two related functions. The first Chebyshev function is given by

with the sum extending over all prime numbers p that are less than x. The second Chebyshev function ψ(x) is defined by

where Λ is the von Mangoldt function. The Chebyshev function is often used in proofs related to prime numbers, because it is typically simpler to work with than the prime counting function, π(x). The von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. ... In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are 1 and itself. ... In mathematics, the prime counting function is the function counting the number of primes less than or equal to some real number x. ...

Both functions are named in honour of Pafnuty Lvovich Chebyshev. Pafnuty Lvovich Chebyshev Pafnuty Lvovich Chebyshev (Пафнутий Львович Чебышёв) (May 4, 1821 - November 26, 1894) was a Russian mathematician. ...

Relationships

The second Chebyshev function can be seen to be related to the first by writing it as

where k is the unique integer such that but p^{k + 1} > x. A more direct relationship is given by

Note that this last sum has only a finite number of non-vanishing terms, as

for n > log₂x.

The exact formula

In 1895, Hans Carl Friedrich von Mangoldt proved^[1] an explicit expression for ψ(x) as a sum over the nontrivial zeros of the Riemann zeta function: Hans Carl Friedrich von Mangoldt (1854-1925) was a German mathematician who contributed to the solution of the prime number theorem. ... In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...

Here ρ runs over the nontrivial zeros of the zeta function, and

From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of − x^ω / ω over the trivial zeros of the zeta function, , i.e. As the degree of the Taylor series rises, it approaches the correct function. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

Properties

A theorem due to Erhard Schmidt states that, for any real, positive K, there are values of x such that Erhard Schmidt (January 13, 1876 - December 6, 1959) was a German mathematician born in Dorpat (now Tartu, Estonia). ...

and

infinitely often^[2][3]. On big-O notation, one may write the above as The Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. ...

Hardy and Littlewood^[4] prove the stronger result, that

Relation to the prime counting function

The Chebyshev function can be related to the prime counting function as follows. Define

Then

.

The transition from π₁ to the prime counting function, π, is made through the equation

Certainly , so for the sake of approximation, this last relation can be recast in the form

.

The Riemann hypothesis

The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part 1/2. In this case, , and it can be shown that Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ...

.

By the above, this implies

Smoothing function

The smoothed Chebyshev function ψ₁(x) − x² / 2, for x < 10⁶

The smoothing function is defined as

It can be shown that

Variational formulation

The Chebyshev function evaluated at x = exp(t) minimizes the functional

so

for c > 0.

References

• ↑ Erhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", Mathematische Annalen, 57 (1903), pp.195-204.
• ↑ G.H. Hardy and J.E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41(1916) pp.119-196.
• ↑ Davenport, Harold (2000). In Multiplicative Number Theory. Springer. p. 104. ISBN 0-387-95097-4. Google Book Search.
• Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9
• Eric W. Weisstein, Chebyshev functions at MathWorld.
• Mangoldt summatory function on PlanetMath
• Chebyshev functions on PlanetMath
• Riemann's Explicit Formula, with images and movies