Von Mangoldt function explained

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

Λ (n) = \begin{cases} log p & if n = p ^{k} forsomeprime p andinteger k \ge 1 , \ 0 & 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 .

Properties

The von Mangoldt function satisfies the identity^{[1] [2]}

log (n) = \sum _{d} Λ (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} Λ (d) & = Λ ( 1 ) + Λ ( 2 ) + Λ ( 3 ) + Λ ( 4 ) + Λ ( 6 ) + Λ ( 12 ) \\ & = Λ ( 1 ) + Λ ( 2 ) + Λ ( 3 ) + Λ \left ( 2 ^{2} \right ) + Λ ( 2 x 3 ) + Λ \left ( 2 ^{2} x 3 \right) \\ & = 0 + log ( 2 ) + log ( 3 ) + log ( 2 ) + 0 + 0 \\ & =log ( 2 x 3 x 2 ) \\ & = log ( 12 ). \end{align}

By Möbius inversion, we have

Λ (n) = \sum _{d} \mu(d) log \left( (n)/(d) \right) and using the product rule for the logarithm we get^{[2] [3] [4]} Λ (n) = - \sum _{d} \mu(d) log (d) .

For all

x\ge 1 , we have^[5] \sum _{n\le} (Λ (n))/(n) =log x + O( 1 ). Also, there exist positive constants and such that\psi(x)\le c _{1x,} for all x\ge 1 , and\psi(x)\ge c _{2x,} for all sufficiently large .

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 ^{inf ty} _{n =2} (Λ (n))/(log (n)) (1)/(n ^{s}) , Re (s) > 1 .

The logarithmic derivative is then^[6]

(\zeta ^{\prime(s)})/(\zeta(s)) = - \sum ^{inf ty} _{n =1} (Λ (n))/(n ^{s}) .

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

F(s) = \sum ^{inf ty} _{n =1} (f(n))/(n ^{s})

for a completely multiplicative function, and the series converges for, then

(F ^{\prime(s)})/(F(s)) = - \sum ^{inf ty} _{n =1} (f(n) Λ (n))/(n ^{s})

converges for .

Chebyshev function

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

\psi(x) = \sum _{p ^{k\le} x} log p = \sum _{n} Λ (n) .

It was introduced by Pafnuty Chebyshev who used it to show that the true order of the prime counting function

\pi(x) is x/ log x . 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.

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

(\zeta ^{\prime(s)})/(\zeta(s)) = - s\int ^{inf ty} _{1} (\psi(x))/(x ^{s+1}) dx

which holds for .

Exponential series

Hardy and Littlewood examined the series^[8]

F(y) = \sum ^{inf ty} _{n =2} \left( Λ (n) -1 \right) e ^{-ny}

in the limit . Assuming the Riemann hypothesis, they demonstrate that

F(y) = O\left( (1)/(\sqrt{y) }\right)\quad \text\quad F(y)=\Omega_\pm\left(\frac\right)

In particular this function is oscillatory with diverging oscillations: there exists a value such that both inequalities

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

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} \left( 1- (n)/(λ) \right) ^{\delta} Λ (n) & =- (1)/(2 \pi i) \int ^{c + i inf ty} _{c - i inf ty} (\Gamma( 1+ \delta)\Gamma(s))/(\Gamma( 1+ \delta + s)) (\zeta ^{\prime(s)})/(\zeta(s)) λ ^{s} ds \\ & = (λ)/(1+ \delta) + \sum _{\rho} (\Gamma( 1+ \delta)\Gamma(\rho))/(\Gamma( 1+ \delta + \rho)) + \sum _{n} c _{n} λ ^{-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} λ ^{-n}

can be shown to be a convergent series for .

Approximation by Riemann zeta zeros

There is an explicit formula for the summatory Mangoldt function

\psi(x) given by^[9] \psi(x) = x - \sum _{\zeta(\rho)=0} (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, 0<\Re(\rho)<1} (x ^{\rho})/(-log ( 2 \pi) - \rho) (12log ( 1- x)/(^{-2}) ). (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]

\sum _{0<\gamma} t ^{\rho = (- T)/(2 \pi)} Λ (t) + l{O}(log T) (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\lim _{T} (1)/(T) \sum _{0<\gamma} \cos (\alpha log t) =- (Λ (t))/(2 \pi \sqrt{t) }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

The functions

Λ _{k(n)=\sum\limits} _{d\mid} \mu(d) log ^{k(n/d),} where \mu denotes the Möbius function and k denotes a positive integer, generalize the von Mangoldt function. The function Λ _{1} is the ordinary von Mangoldt function Λ .

See also

• Prime-counting function

References

• • Book: Hardy . G. H.. G. H. Hardy . Wright . E. M. . E. M. Wright . Heath-Brown . D. R. . Roger Heath-Brown . Silverman . J. H. . Joseph H. Silverman . An Introduction to the Theory of Numbers . 6th . Oxford University Press . Oxford . 2008 . 1938 . 978-0-19-921985-8. 2445243. 1159.11001. HW.
• Book: Tenebaum, Gérald . C.B. Thomas . 1995 . Introduction to analytic and probabilistic number theory . Cambridge Studies in Advanced Mathematics . 46 . . Cambridge . 0-521-41261-7 . 0831.11001 .

External links

• Allan Gut, Some remarks on the Riemann zeta distribution (2005)
  • Heike, How plot Riemann zeta zero spectrum in Mathematica? (2012)

Notes and References

1. Apostol (1976) p.32
2. Tenenbaum (1995) p.30
3. Apostol (1976) p.33
4. Book: Schroeder, Manfred R. . Number theory in science and communication. With applications in cryptography, physics, digital information, computing, and self-similarity . 3rd . 0997.11501 . Springer Series in Information Sciences . 7 . Berlin . . 1997 . 3-540-62006-0 .
5. Apostol (1976) p.88
6. Hardy & Wright (2008) §17.7, Theorem 294
7. Apostol (1976) p.246
8. G. H. . Hardy . amp . J. E. . Littlewood . Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes . Acta Mathematica . 41 . 1916 . 119–196 . 10.1007/BF02422942 . 2014-07-03 . https://web.archive.org/web/20120207110455/http://www.ift.uni.wroc.pl/%7Emwolf/Hardy_Littlewood%20zeta.pdf . 2012-02-07 . dead . free .
9. Conrey. J. Brian. Brian Conrey. March 2003. The Riemann hypothesis. Notices Am. Math. Soc.. 50. 341–353. 1160.11341. 3. Page 346
10. E. Landau, Über die Nullstellen der Zetafunktion, Math. Annalen 71 (1911), 548-564.