# Von Mangoldt function

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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 Λ(n), is defined as

$\Lambda(n) = \begin{cases} \log p & \mbox{if }n=p^k \mbox{ for some prime } p \mbox{ and integer } k \ge 1, \\ 0 & \mbox{otherwise.} \end{cases}$

The values of Λ(n) for the first nine positive numbers are

$\log 1 , \log 2 , \log 3 , \log 2 , \log 5 , \log 1 , \log 7 , \log 2 , \log 3,$

which is related to (sequence A014963 in OEIS).

The summatory von Mangoldt function, ψ(x), also known as the Chebyshev function, is defined as

$\psi(x) = \sum_{n\le x} \Lambda(n).$

von Mangoldt provided a rigorous proof of an explicit formula for ψ(x) 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 identity[1][2]

$\log(n) = \sum_{d | n} \Lambda(d).$

The sum is taken over all integers d that divide n. This is proved by the fundamental theorem of arithmetic, since the terms that are not powers of primes are equal to 0. For example, consider the case n = 12 = 22 × 3. Then

\begin{align} \sum_{d | 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 have[2][3][4]

$\Lambda (n) = - \sum_{d | n} \mu(d) \log(d) \ .$

## Dirichlet series

The von Mangoldt function plays an important role in the theory of Dirichlet series, and in particular, the Riemann zeta function. In particular, 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 then

$\frac {\zeta^\prime(s)}{\zeta(s)} = -\sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}.$

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 f (n), and the series converges for Re(s) > σ0, then

$\frac {F^\prime(s)}{F(s)} = - \sum_{n=1}^\infty \frac{f(n)\Lambda(n)}{n^s}$

converges for Re(s) > σ0.

## Chebyshev function

The second Chebyshev function ψ(x) is the summatory function of the von Mangoldt function:[5]

$\psi(x) = \sum_{p^k\le x}\log p=\sum_{n \leq x} \Lambda(n) \ .$

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 Re(s) > 1.

## Exponential series

Hardy and Littlewood examine the series[6]

$F(y)=\sum_{n=2}^\infty \left(\Lambda(n)-1\right) e^{-ny}$

in the limit y → 0+. Assuming the Riemann hypothesis, they demonstrate that

$F(y)=O\left(\sqrt{\frac{1}{y}}\right).$

Curiously, they also show that this function is oscillatory as well, with diverging oscillations. In particular, there exists a value K > 0 such that

$F(y)< -\frac{K}{\sqrt{y}}, \quad \text{ and } \quad F(y)> \frac{K}{\sqrt{y}}$

infinitely often. 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 y < 10−5.

## 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, λ and δ are numbers characterizing the Riesz mean. One must take c > 1. The sum over ρ 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 λ > 1.

## Approximation by Riemann zeta zeros

The first Riemann zeta zero wave in the sum that approximates the von Mangoldt function

The real part of the sum over the zeta zeros:[citation needed]

$\Lambda(n) = -\sum_{i=1}^{\infty} n^{\rho(i)}$

where ρ(i) is the i-th zeta zero, approximates the von Mangoldt function by summing several waves onto each other.[7]

The Fourier transform of the von Mangoldt function gives a spectrum with imaginary parts of Riemann zeta zeros as spikes at the x-axis ordinates (right), while the von Mangoldt function can be approximated by Zeta zero waves (left)

The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinates equal to imaginary part of the Riemann zeta function zeros. This is sometimes called a duality.

## References

1. ^ Apostol (1976) p.32
2. ^ a b Tenenbaum (1995) p.30
3. ^ Apostol (1976) p.33
4. ^ Schroeder, Manfred R. (1997). Number theory in science and communication. With applications in cryptography, physics, digital information, computing, and self-similarity. Springer Series in Information Sciences 7 (3rd ed.). Berlin: Springer-Verlag. ISBN 3-540-62006-0. Zbl 0997.11501.
5. ^ Apostol (1976) p.246
6. ^ Hardy, G. H. & Littlewood, J. E. (1916). "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes". Acta Mathematica 41: 119–196. doi:10.1007/BF02422942.
7. ^ Conrey, J. Brian (March 2003). "The Riemann hypothesis". Notices Am. Math. Soc. 50 (3): 341–353. Zbl 1160.11341. Page 346