Von Mangoldt function
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 numbers are
The summatory von Mangoldt function, ψ(x), also known as the Chebyshev function, is defined as
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 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
The logarithmic derivative is then
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.
which holds for Re(s) > 1.
in the limit y → 0+. Assuming the Riemann hypothesis, they demonstrate that
Curiously, they also show that this function is oscillatory as well, with diverging oscillations. In particular, there exists a value K > 0 such that
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.
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
The real part of the sum over the zeta zeros:
- , where ρ(i) is the i-th zeta zero, peaks at primes, as can be seen in the adjoining graph, and can also be verified through numerical computation. It does not sum up to the Von Mangoldt function.
The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinates equal to imaginary part of the Riemann zeta function zeros. This is sometimes called a duality.
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