Von Mangoldt function

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

1 Definition
2 Dirichlet series
3 Mellin transform
4 Exponential series
5 Riesz mean
6 See also
7 References

[edit] Definition

The von Mangoldt function, conventionally written as Λ(n), is defined as

$\Lambda(n) = \begin{cases} \log p & \mbox{if }n=p^k \mbox{ for some prime } p \mbox{ and integer } k \ge 1, \\ 0 & \mbox{otherwise.} \end{cases}$

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 which divide 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.

[edit] 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.^[2] 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 $\Re(s) > \sigma_0$ .

[edit] 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 $\Re(s)>1$ .

[edit] Exponential series

An exponential series involving the von Mangoldt function, summed up to the first

109

terms

Hardy and Littlewood examine the series^[3]

$F(y)=\sum_{n=2}^\infty \left(\Lambda(n)-1\right) e^{-ny}$

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 $y < 10 - 5$ .

[edit] 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, $λ$ 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

∑	c_nλ ^{− n}
n

can be shown to be a convergent series for $λ > 1$ .

[edit] See also

Prime counting function

[edit] References

↑ Allan Gut, Some remarks on the Riemann zeta distribution (2005)
↑ Tom Apostol, Introduction to analytic number theory, Springer-Verlag, New York, 1976. (See theorem 2.10)
↑ G.H. Hardy and J.E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41(1916) pp.119-196.
S.A. Stepanov, "Mangoldt function" SpringerLink Encyclopaedia of Mathematics (2001)