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Wikipedia / 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

, is defined as

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

is the Möbius function. Then by using the product rule for the logarithm we get

, we have

Also, there exist positive constants and such that

, 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 is the summatory function of the von Mangoldt function:

It was introduced by Pafnuty Chebyshev 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:

.

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

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.)

In the opposite direction, in 1911 E. Landau proved that for any fixed t > 1

(We use the notation ρ = β + iγ for the non-trivial zeros of the zeta function.) Therefore, if we use Riemann notation α = −i(ρ − 1/2) we have that the sum over nontrivial zeta zeros expressed as

peaks at primes and powers of primes.

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

denotes the Möbius function denotes a positive integer, generalize the von Mangoldt function..

See also

References