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Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt.
Definition
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]
\( \log n = \sum_{d\,\mid\,n} \Lambda(d),\, \)
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.
For instance, let n=12. Recall the prime factorization of 12, 12=22·3, which will turn up in the example.
Take the summation over all distinct positive divisors d of n:
\( \sum_{d\,\mid\,12} \Lambda(d) = \Lambda(1) + \Lambda(2) + \Lambda(3) + \Lambda(4) + \Lambda(6) + \Lambda(12) \)
\( = \Lambda(1) + \Lambda(2) + \Lambda(3) + \Lambda(2^2) + \Lambda(2 \times 3) + \Lambda(2^2 \times 3) \)
\( = 0 + \log 2 + \log 3 + \log 2 + 0 + 0 \,\! \)
\( =\log (2 \times 3 \times 2) = \log 12. \,\! \)
This provides an example of how the summation of the von Mangoldt function equals log (n).
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
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
\( \log \zeta(s)=\sum_{n=2}^\infty \frac{\Lambda(n)}{\log(n)}\,\frac{1}{n^s} \)
for \( \Re(s) > 1 \). 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) > \sigma_0, then
\( \frac {F^\prime(s)}{F(s)} = - \sum_{n=1}^\infty \frac{f(n)\Lambda(n)}{n^s} \)
converges for \( \Re(s) > \sigma_0. \)
Mellin transform
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
An exponential series involving the von Mangoldt function, summed up to the first 10^9 terms
Hardy and Littlewood examine the series[2]
\( F(y)=\sum_{n=2}^\infty \left(\Lambda(n)-1\right) e^{-ny} \)
in the limit y\to 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
\( F(y)< -\frac{K}{\sqrt{y}} and F(y)> \frac{K}{\sqrt{y}} \)
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
\( \sum_{n\le \lambda} \left(1-\frac{n}{\lambda}\right)^\delta \Lambda(n) = - \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{\Gamma(1+\delta)\Gamma(s)}{\Gamma(1+\delta+s)} \frac{\zeta^\prime(s)}{\zeta(s)} \lambda^s ds \)
\( = \frac{\lambda}{1+\delta} + \sum_\rho \frac {\Gamma(1+\delta)\Gamma(\rho)}{\Gamma(1+\delta+\rho)} +\sum_n c_n \lambda^{-n}. \)
Here, \( \lambda \) and \( \delta \) are numbers characterizing the Riesz mean. One must take c>1. The sum over \rho 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 \( \lambda > 1. \)
Expansion of terms
The terms of the von Mangoldt function can be expanded into series which have numerators that form an infinite Greatest common divisor pattern symmetric matrix starting:
\( T = \begin{bmatrix} 1&1&1&1&1&1 \\ 1&-1&1&-1&1&-1 \\ 1&1&-2&1&1&-2 \\ 1&-1&1&-1&1&-1 \\ 1&1&1&1&-4&1 \\ 1&-1&-2&-1&1&2 \end{bmatrix} \)
This matrix [3] is defined by the recurrence:
\( T(n,1)=1,\;T(1,k)=1,\;n \geq k: T(n,k) = -\sum\limits_{i=1}^{k-1} T(n-i,k),\;n<k: T(n,k) = -\sum\limits_{i=1}^{n-1} T(k-i,n) \)
or:
\( T(n,k) = a(GCD(n,k)) \)
where "a" is the Dirichlet inverse of the Euler's totient function.
The von Mangoldt function can then for n>1 be calculated as:[4]
\( \Lambda(n) = \sum\limits_{k=1}^{\infty}\frac{T(n,k)}{k} \)
Or:
\( \Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(s-1)}} \)
where
\( \zeta(s) \) is the Riemann zeta function, \mu is the Möbius function and d is a divisor.
Approximation by Riemann zeta zeros
The real part of the sum over the zeta zeros:
\( \Lambda(n) = -\sum\limits_{i=1}^{\infty} n^{\rho(i)} \)
where
\( \rho(i) is the i-th zeta zero, \)
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)
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