Von Mangoldt function
The von Mangoldt function, conventionally written as Λ(n), is defined as
which is a sequence starting:
The von Mangoldt function satisfies the identity
- 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:
- 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
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.
for . 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 , and the series converges for , then
converges for .
which holds for .
Hardy and Littlewood examine the series
in the limit . 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 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 .
The Riesz mean of the von Mangoldt function is given by
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
can be shown to be a convergent series for .
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:
This matrix  is defined by the recurrence:
where "a" is the Dirichlet inverse of the Euler's totient function, and for :
The von Mangoldt function can then for be calculated as:
Approximation by Riemann zeta zeros
The real part of the sum over the zeta zeros:
- is the i-th zeta zero,
approximates the von Mangoldt function by summing several waves onto each other (page 346).
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 exponentiated von Mangoldt function can be described by a recurrence in a two dimensional matrix.
- 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
- 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.
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