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).
Properties
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The von Mangoldt function satisfies the identity[1][2]
The sum is taken over all integersd that dividen. 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
Also, there exist positive constants c1 and c2 such that
for all , and
for all sufficiently large x.
Dirichlet series
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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
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.
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
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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
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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.)
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
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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 .
^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.
^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.
^Conrey, J. Brian (March 2003). "The Riemann hypothesis" (PDF). Notices Am. Math. Soc. 50 (3): 341–353. Zbl 1160.11341. Page 346
^E. Landau, Über die Nullstellen der Zetafunktion, Math. Annalen 71 (1911 ), 548-564.
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. Vol. 46. Translated by C.B. Thomas. Cambridge: Cambridge University Press. ISBN 0-521-41261-7. Zbl 0831.11001.
External links
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Allan Gut, Some remarks on the Riemann zeta distribution (2005)