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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.


The von Mangoldt function, denoted by Λ(n), is defined as

The values of Λ(n) for the first nine positive integers (i.e. natural numbers) are

which is related to (sequence A014963 in the OEIS).


The von Mangoldt function satisfies the identity[1][2]

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

By Möbius inversion, we have

and using the product rule for the logarithm we get[2][3][4]

For all , we have[5]

Also, there exist positive constants c1 and c2 such that

for all , and

for all sufficiently large x.

Dirichlet series[edit]

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[6]

These are special cases of a more general relation on Dirichlet series. If one has

for a completely multiplicative function f(n), and the series converges for Re(s) > σ0, then

converges for Re(s) > σ0.

Chebyshev function[edit]

The second Chebyshev function ψ(x) is the summatory function of the von Mangoldt function:[7]

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 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.

The Mellin transform of the Chebyshev function can be found by applying Perron's formula:

which holds for Re(s) > 1.

Exponential series[edit]

Hardy and Littlewood examined the series[8]

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

In particular this function is oscillatory with diverging oscillations: there exists a value K > 0 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 y < 10−5.

Riesz mean[edit]

The Riesz mean of the von Mangoldt function is given by

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

can be shown to be a convergent series for λ > 1.

Approximation by Riemann zeta zeros[edit]

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 given by[9]

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[10]

(We use the notation ρ = β + iγ for the non-trivial zeros of the zeta function.)

(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.

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[edit]

The functions

where denotes the Möbius function and denotes a positive integer, generalize the von Mangoldt function.[11] The function is the ordinary von Mangoldt function .

See also[edit]


  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. Vol. 7 (3rd ed.). Berlin: Springer-Verlag. ISBN 3-540-62006-0. Zbl 0997.11501.
  5. ^ Apostol (1976) p.88
  6. ^ Hardy & Wright (2008) §17.7, Theorem 294
  7. ^ Apostol (1976) p.246
  8. ^ 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. Archived from the original (PDF) on 2012-02-07. Retrieved 2014-07-03.
  9. ^ Conrey, J. Brian (March 2003). "The Riemann hypothesis" (PDF). Notices Am. Math. Soc. 50 (3): 341–353. Zbl 1160.11341. Page 346
  10. ^ E. Landau, Über die Nullstellen der Zetafunktion, Math. Annalen 71 (1911 ), 548-564.
  11. ^ Iwaniec, Henryk; Friedlander, John (2010), Opera de cribro, American Mathematical Society Colloquium Publications, vol. 57, Providence, RI: American Mathematical Society, p. 23, ISBN 978-0-8218-4970-5, MR 2647984

External links[edit]