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
\Lambda(n) = \begin{cases} \log p & \text{if }n=p^k \text{ for some prime } p \text{ and integer } k \ge 1, \\ 0 & \text{otherwise.} \end{cases}
The values of  for the first nine positive integers (i.e. natural numbers) are
0 , \log 2 , \log 3 , \log 2 , \log 5 , 0 , \log 7 , \log 2 , \log 3,
which is related to .
The summatory von Mangoldt function, , also known as the second Chebyshev function, is defined as
\psi(x) = \sum_{n\le x} \Lambda(n).
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.
Properties
The von Mangoldt function satisfies the identityApostol (1976) p.32Tenenbaum (1995) p.30
\log(n) = \sum_{d \mid n} \Lambda(d).
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
\begin{align} \sum_{d \mid 12} \Lambda(d) &= \Lambda(1) + \Lambda(2) + \Lambda(3) + \Lambda(4) + \Lambda(6) + \Lambda(12) \\ &= \Lambda(1) + \Lambda(2) + \Lambda(3) + \Lambda \left (2^2 \right ) + \Lambda(2 \times 3) + \Lambda \left (2^2 \times 3 \right) \\ &= 0 + \log(2) + \log(3) + \log(2) + 0 + 0 \\ &=\log (2 \times 3 \times 2) \\ &= \log(12).
\end{align}
By Möbius inversion, we haveApostol (1976) p.33
\Lambda (n) = - \sum_{d \mid n} \mu(d) \log(d) \ .
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
\log \zeta(s)=\sum_{n=2}^\infty \frac{\Lambda(n)}{\log(n)}\,\frac{1}{n^s}, \qquad \text{Re}(s) > 1.
The logarithmic derivative is thenHardy & Wright (2008) §17.7, Theorem 294
\frac {\zeta^\prime(s)}{\zeta(s)} = -\sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}.
These are special cases of a more general relation on Dirichlet series.
If one has
F(s) =\sum_{n=1}^\infty \frac{f(n)}{n^s}
for a completely multiplicative function , and the series converges for , then
\frac {F^\prime(s)}{F(s)} = - \sum_{n=1}^\infty \frac{f(n)\Lambda(n)}{n^s}
converges for .
Chebyshev function
The second Chebyshev function ψ(x) is the summatory function of the von Mangoldt function:Apostol (1976) p.246
\psi(x) = \sum_{p^k\le x}\log p=\sum_{n \leq x} \Lambda(n) \ .
The Mellin transform of the Chebyshev function can be found by applying Perron's formula:
\frac{\zeta^\prime(s)}{\zeta(s)} = - s\int_1^\infty \frac{\psi(x)}{x^{s+1}}\,dx
which holds for .
Exponential series
thumb|right|400px
Hardy and Littlewood examined the series
F(y)=\sum_{n=2}^\infty \left(\Lambda(n)-1\right) e^{-ny}
in the limit .
Assuming the Riemann hypothesis, they demonstrate that
F(y)=O\left(\frac{1}{\sqrt{y}}\right)\quad \text{and}\quad F(y)=\Omega_\pm\left(\frac{1}{\sqrt{y}}\right)
In particular this function is oscillatory with diverging oscillations: there exists a value  such that both inequalities
F(y)< -\frac{K}{\sqrt{y}}, \quad \text{ and } \quad F(z)> \frac{K}{\sqrt{z}}
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
\begin{align} \sum_{n\le \lambda} \left(1-\frac{n}{\lambda}\right)^\delta \Lambda(n) &= -\frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty}  \frac{\Gamma(1+\delta)\Gamma(s)}{\Gamma(1+\delta+s)} \frac{\zeta^\prime(s)}{\zeta(s)} \lambda^s ds \\ &= \frac{\lambda}{1+\delta} + \sum_\rho \frac{\Gamma(1+\delta)\Gamma(\rho)}{\Gamma(1+\delta+\rho)} + \sum_n c_n \lambda^{-n}.
\end{align}
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
\sum_n c_n \lambda^{-n}\,
can be shown to be a convergent series for .
Approximation by Riemann zeta zeros
thumb|The first Riemann zeta zero wave in the sum that approximates the von Mangoldt function
There is an explicit formula for the summatory Mangoldt function \psi(x) given by  Page 346
\psi(x)=x-\sum_{\zeta(\rho)=0}\frac{x^\rho}\rho -\log(2\pi).
If we separate out the trivial zeros of the zeta function, which are the negative even integers, we obtain
\psi(x)=x-\sum_{\zeta(\rho)=0,\ \Re(\rho)>0}\frac{x^\rho}\rho -\log(2\pi)-\frac12\log(1-x^{-2}).
Taking the derivative of both sides, ignoring convergence issues, we get an "equality" of distributions
\sum_{q=p^r}\Lambda(q)\delta(x-q)= 1-\sum_{\zeta(\rho)=0,\ \Re(\rho)>0}\frac{x^\rho}x+\frac{1}{x-x^3}.
(Left) The von Mangoldt function, approximated by zeta zero waves.(Right)
The Fourier transform of the von Mangoldt function gives a spectrum with imaginary parts of Riemann zeta zeros as spikes at the -axis ordinates.
|thumb Therefore, we should expect that the sum over nontrivial zeta zeros
-\sum_{\zeta(\rho)=0,\ \Re(\rho)>0}\frac{x^{\rho}}{x}
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.
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)
