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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

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

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

$$0 , \log 2 , \log 3 , \log 2 , \log 5 , 0 , \log 7 , \log 2 , \log 3,$$

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]

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

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 \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 have[2][3][4]

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

Dirichlet series

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

Mangoldt-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

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

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:

$$-\sum_{i=1}^{\infty} n^{\rho(i)}$$ , where ρ(i) is the i-th zeta zero, peaks at primes, as can be seen in the adjoining graph, and can also be verified through numerical computation. It does not sum up to the Von Mangoldt function.[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.

Prime-counting function

References

Apostol (1976) p.32
Tenenbaum (1995) p.30
Apostol (1976) p.33
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.
Apostol (1976) p.246
Hardy, G. H. & Littlewood, J. E. (1916). "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes" (PDF). Acta Mathematica 41: 119–196. doi:10.1007/BF02422942.

Conrey, J. Brian (March 2003). "The Riemann hypothesis" (PDF). Notices Am. Math. Soc. 50 (3): 341–353. Zbl 1160.11341. Page 346

Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
Tenebaum, Gérald (1995). Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics 46. Translated by C.B. Thomas. Cambridge: Cambridge University Press. ISBN 0-521-41261-7. Zbl 0831.11001.

Allan Gut, Some remarks on the Riemann zeta distribution (2005)
S.A. Stepanov (2001), "Mangoldt function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Chris King, Primes out of thin air (2010)
Heike, How plot Riemann zeta zero spectrum in Mathematica? (2012)