Von Mangoldt function

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In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt.

Definition[edit]

The von Mangoldt function, conventionally written as 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}

which is a sequence starting:

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

oeis sequence A014963

It is an example of an important arithmetic function that is neither multiplicative nor additive.

The von Mangoldt function satisfies the identity[1]

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

that is, 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.

We provide an example of how the summation of the von Mangoldt function equals log(n). Let n = 12 = 22 × 3, then we can write:
\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}

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.

Dirichlet series[edit]

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.[1] 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.

Mellin transform[edit]

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[edit]

Hardy and Littlewood examine the series[2]

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)=\mathcal{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[edit]

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[edit]

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 [1](page 346).

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

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

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

See also[edit]

Notes[edit]

  1. ^ a b 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 
  2. ^ 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. 

References[edit]