In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the von Mangoldt function is an
arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...
named after
German
German(s) may refer to:
* Germany (of or related to)
** Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
mathematician
Hans von Mangoldt. It is an example of an important arithmetic function that is neither
multiplicative nor
additive
Additive may refer to:
Mathematics
* Additive function, a function in number theory
* Additive map, a function that preserves the addition operation
* Additive set-functionn see Sigma additivity
* Additive category, a preadditive category with f ...
.
Definition
The von Mangoldt function, denoted by , is defined as
:
The values of for the first nine positive integers (i.e. natural numbers) are
:
which is related to .
Properties
The von Mangoldt function satisfies the identity
[Apostol (1976) p.32][Tenenbaum (1995) p.30]
:
The sum is taken over all
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s that
divide . This is proved by the
fundamental theorem of arithmetic
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
, since the terms that are not powers of primes are equal to . For example, consider the case . Then
:
By
Möbius inversion
Moebius, Möbius or Mobius may refer to:
People
* August Ferdinand Möbius (1790–1868), German mathematician and astronomer
* Theodor Möbius (1821–1890), German philologist
* Karl Möbius (1825–1908), German zoologist and ecologist
* Paul ...
, we have
:
and using the product rule for the logarithm we get
[Apostol (1976) p.33]
:
For all , we have[Apostol (1976) p.88]
:
Also, there exist positive constants and such that
:
for all , and
:
for all sufficiently large .
Dirichlet series
The von Mangoldt function plays an important role in the theory of Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analy ...
, and in particular, the Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
. For example, one has
:
The logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula
\frac
where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f'' ...
is then[Hardy & Wright (2008) §17.7, Theorem 294]
:
These are special cases of a more general relation on Dirichlet series. If one has
:
for a completely multiplicative function In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime ...
, and the series converges for , then
:
converges for .
Chebyshev function
The second Chebyshev function
In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function or is given by
:\vartheta(x)=\sum_ \ln p
where \ln denotes the natural logarithm, ...
''ψ''(''x'') is the summatory function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...
of the von Mangoldt function:[Apostol (1976) p.246]
:
It was introduced by Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebyshe ...
who used it to show that the true order of the prime counting function is . Von Mangoldt provided a rigorous proof of an explicit formula for involving a sum over the non-trivial zeros of the Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
. This was an important part of the first proof of the prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
.
The Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used i ...
of the Chebyshev function can be found by applying Perron's formula In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transform.
Statement
Let \ be an arithmetic function, a ...
:
:
which holds for .
Exponential series
Hardy and Littlewood examined the series
:
in the limit . Assuming the Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
, they demonstrate that
:
In particular this function is oscillatory with diverging oscillations
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
: there exists a value such that both inequalities
:
hold infinitely often in any neighbourhood of 0. The graphic to the right indicates that this behaviour is not at first numerically obvious: the oscillations are not clearly seen until the series is summed in excess of 100 million terms, and are only readily visible when .
Riesz mean
The Riesz mean of the von Mangoldt function is given by
:
Here, and are numbers characterizing the Riesz mean. One must take . The sum over is the sum over the zeroes of the Riemann zeta function, and
:
can be shown to be a convergent series for .
Approximation by Riemann zeta zeros
There is an explicit formula for the summatory Mangoldt function given by
:
If we separate out the trivial zeros of the zeta function, which are the negative even integers, we obtain
:
Taking the derivative of both sides, ignoring convergence issues, we get an "equality" of distributions
:
Therefore, we should expect that the sum over nontrivial zeta zeros
:
peaks at primes. In fact, this is the case, as can be seen in the adjoining graph, and can also be verified through numerical computation.
The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinates equal to the imaginary parts of the Riemann zeta function zeros. This is sometimes called a duality.
Generalized von Mangoldt function
The functions
:
where denotes the Möbius function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most of ...
and denotes a positive integer, generalize the von Mangoldt function. The function is the ordinary von Mangoldt function .
See also
* Prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ).
History
Of great interest in number theory is t ...
References
*
*
*
External links
* Allan Gut,
Some remarks on the Riemann zeta distribution
' (2005)
* {{springer, id=m/m062200, author=S.A. Stepanov, title=Mangoldt function
* Heike,
How plot Riemann zeta zero spectrum in Mathematica?
' (2012)
Arithmetic functions