Von Mangoldt function

Von Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
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
The sum is taken over all integers that divide . This is proved by the fundamental theorem of arithmetic, since the terms that are not powers of primes are equal to . For example, consider the case . Then
By Möbius inversion, we have
and using the product rule for the logarithm we get
For all , we have
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, and in particular, the Riemann zeta function. For example, one has
The logarithmic derivative is then
These are special cases of a more general relation on Dirichlet series. If one has
for a completely multiplicative function , and the series converges for , then
converges for .
Chebyshev function
The second Chebyshev function ψ(x) is the summatory function of the von Mangoldt function:
It was introduced by Pafnuty Chebyshev 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. This was an important part of the first proof of the prime number theorem.
The Mellin transform of the Chebyshev function can be found by applying Perron's formula:
which holds for .
Exponential series
Hardy and Littlewood examined the series
in the limit . Assuming the Riemann hypothesis, they demonstrate that
In particular this function is oscillatory with diverging oscillations: 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
(The sum is not absolutely convergent, so we take the zeros in order of the absolute value of their imaginary part.)
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 and denotes a positive integer, generalize the von Mangoldt function. The function is the ordinary von Mangoldt function .
See also
Prime-counting function
References
External links
Allan Gut, Some remarks on the Riemann zeta distribution (2005)
Heike, How plot Riemann zeta zero spectrum in Mathematica? (2012)
Arithmetic functions